# Công thức giải tích

Buớc tưới chuyển hướng Bước tới tìm kiếm

## Biến đổi hàm số

Biến đổi hàm số cho biết tỉ lệ thay đổi của hàm số trên thay đổi biến số

${\displaystyle {\frac {\Delta f(x)}{\Delta x}}={\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$

Biến đổi hàm số được dùng để cho biết độ dóc của một hình có thể biểu diển bằng một hàm số f(x)

${\displaystyle \Delta f(x)=a\Delta x}$

Từ trên,

${\displaystyle f(x+\Delta x)-f(x)=a\Delta x}$
${\displaystyle f(x+\Delta x)=a\Delta x+f(x)}$

## Limit

Phép toán giải tích tìm giá trị của một hàm số khi biến số của hàm số tiến tới một trị số . Ký hiệu toán limit ${\displaystyle \lim _{\Delta x\to 0}f(x)}$ . Phép toán Limit được thực hiện như sau

${\displaystyle \lim _{\Delta x\to 0}f(x)=L}$

Luật toán giải tích Limit

 Hàm số Limit ${\displaystyle f(x)\pm g(x)}$ ${\displaystyle \lim _{x\to c}f(x)\pm \lim _{x\to c}g(x)}$ ${\displaystyle f(x)\times g(x)}$ ${\displaystyle \lim _{x\to c}f(x)\times \lim _{x\to c}g(x)}$ ${\displaystyle kf(x)}$ ${\displaystyle k\lim _{x\to c}f(x)}$ ${\displaystyle n{\sqrt {f(x)}}}$ ${\displaystyle n{\sqrt {\lim _{x\to c}f(x)}}}$

Thí dụ

${\displaystyle \lim _{x\to 0}x=0}$
${\displaystyle \lim _{x\to 0}{\frac {1}{x}}=00}$
${\displaystyle \lim _{x\to 00}x=00}$
${\displaystyle \lim _{x\to 00}{\frac {1}{x}}=0}$

## Tổng dải số

1 + 2 + 3 + 4 + 5 + ... + 10 = (1+10) + (2+9) + (3+8) + (4+7) + (5+6) = 11 x 5 = 55
${\displaystyle \sum _{n=1}^{n}m=(1+m)}$ x (# pairs)
 Tổng dải số Ký hiệu Giá trị 1 + 2 + 3 + 4 + .... ${\displaystyle \sum _{n=1}^{\infty }n}$ ${\displaystyle 1+{1 \over 2}+{1 \over 4}+{1 \over 8}+{1 \over 16}+\cdots }$ ${\displaystyle \sum _{n=0}^{\infty }{1 \over 2^{n}}}$ ${\displaystyle 1+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+\cdots }$ ${\displaystyle \sum _{n=1}^{\infty }{1 \over n}}$ ${\displaystyle 1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots }$ ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}{1 \over n}}$

## Đạo hàm

 Ký hiệu Giá trị ${\displaystyle {\frac {df(x)}{dx}}=f^{'}(x)}$ ${\displaystyle \lim _{\Delta x\to 0}\sum {\frac {\Delta f(x)}{\Delta x}}=\lim _{\Delta x\to 0}\sum {\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$ ${\displaystyle \lim _{h\to 0}\sum {\frac {\Delta f(h)}{h}}=\lim _{h\to 0}\sum {\frac {f(x+h)-f(x)}{h}}}$
 Hàm số lũy thừa Đạo hàm hàm số ${\displaystyle e^{x}}$ ${\displaystyle e^{x}}$ ${\displaystyle n^{x}}$ ${\displaystyle n^{x}Lnn}$ ${\displaystyle x^{n}}$ ${\displaystyle nx^{n-1}}$ ${\displaystyle x^{1}}$ ${\displaystyle x}$ ${\displaystyle c}$ ${\displaystyle 0}$ ${\displaystyle Lnx}$ ${\displaystyle {\frac {1}{x}}}$ Hàm số lượng giác thuận Đạo hàm hàm số ${\displaystyle \cos x}$ ${\displaystyle -\sin x}$ ${\displaystyle \sin x}$ ${\displaystyle \cos x}$ ${\displaystyle \tan x}$ ${\displaystyle \sec ^{2}x}$ ${\displaystyle \cot x}$ ${\displaystyle -\csc ^{2}x}$ ${\displaystyle \sec x}$ ${\displaystyle \sec x\tan x}$ ${\displaystyle \csc x}$ ${\displaystyle -\csc x\cot x}$ Hàm số lượng giác nghịch Đạo hàm hàm số ${\displaystyle \cos ^{-1}x}$ ${\displaystyle -{\frac {1}{\sqrt {1-x^{2}}}}}$ ${\displaystyle \sin ^{-1}x}$ ${\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}$ ${\displaystyle \tan ^{-1}x}$ ${\displaystyle {\frac {1}{1+x^{2}}}}$ ${\displaystyle \cot ^{-1}x}$ ${\displaystyle -{\frac {1}{\sqrt {1-x^{2}}}}}$ ${\displaystyle \sec ^{-1}x}$ ${\displaystyle {\frac {1}{x{\sqrt {x^{2}-1}}}}}$ ${\displaystyle \csc ^{-1}x}$ ${\displaystyle -{\frac {1}{x{\sqrt {x^{2}-1}}}}}$ Hàm số đường cong ex thuận Đạo hàm hàm số ${\displaystyle \sinh x}$ ${\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}}$ ${\displaystyle \cosh x}$ ${\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}}$ ${\displaystyle \tanh x}$ ${\displaystyle {\operatorname {sech} ^{2}\,x}}$ ${\displaystyle \operatorname {sech} \,x}$ ${\displaystyle -\tanh x\,\operatorname {sech} \,x}$ ${\displaystyle \operatorname {csch} \,x}$ ${\displaystyle -\,\operatorname {coth} \,x\,\operatorname {csch} \,x}$ ${\displaystyle \operatorname {coth} \,x}$ ${\displaystyle -\,\operatorname {csch} ^{2}\,x}$ Hàm số đường cong ex nghịch Đạo hàm của hàm số ${\displaystyle \operatorname {arsinh} \,x}$ ${\displaystyle {1 \over {\sqrt {x^{2}+1}}}}$ ${\displaystyle \operatorname {arcosh} \,x}$ ${\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}}$ ${\displaystyle \operatorname {artanh} \,x}$ ${\displaystyle {1 \over 1-x^{2}}}$ ${\displaystyle \operatorname {arsech} \,x}$ ${\displaystyle -{1 \over x{\sqrt {1-x^{2}}}}}$ ${\displaystyle \operatorname {arcsch} \,x}$ ${\displaystyle -{1 \over |x|{\sqrt {1+x^{2}}}}}$ ${\displaystyle \operatorname {arcoth} \,x}$ ${\displaystyle {1 \over 1-x^{2}}}$ Đạo hàm hàm số đặc biệt Đạo hàm hàm số ${\displaystyle \Gamma (x)}$ ${\displaystyle \int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt}$ ${\displaystyle \Gamma (x)}$ ${\displaystyle \Gamma (x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n}}\right)-{\dfrac {1}{x+n}}\right)-{\dfrac {1}{x}}\right)=\Gamma (x)\psi (x)}$ ${\displaystyle \zeta (x)}$ ${\displaystyle -\sum _{n=1}^{\infty }{\frac {\ln n}{n^{x}}}=-{\frac {\ln 2}{2^{x}}}-{\frac {\ln 3}{3^{x}}}-{\frac {\ln 4}{4^{x}}}-\cdots \!}$ ${\displaystyle \zeta (x)}$ ${\displaystyle -\sum _{p{\text{ prime}}}{\frac {p^{-x}\ln p}{(1-p^{-x})^{2}}}\prod _{q{\text{ prime}},q\neq p}{\frac {1}{1-q^{-x}}}\!}$ Hàm số Đạo hàm bậc N ${\displaystyle y=F(G(x))\!}$ ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=n!\displaystyle \sum _{\{k_{m}\}}^{}{\dfrac {\mathrm {d} ^{r}}{\mathrm {d} z^{r}}}F(z)|_{z=G(x)}\displaystyle \prod _{m=1}^{n}{\dfrac {1}{k_{m}!}}\left({\dfrac {1}{m!}}{\dfrac {\mathrm {d} ^{m}}{\mathrm {d} x^{m}}}G(x)\right)^{k_{m}}\!}$ where ${\displaystyle r=\displaystyle \sum _{m=1}^{n}k_{m}\!}$ and the set ${\displaystyle \{k_{m}\}\!}$ consists of all non-negative integer solutions of the Diophantine equation ${\displaystyle \displaystyle \sum _{m=1}^{n}mk_{m}=n\!}$ See: Faà di Bruno's formula, Expansions for nearly Gaussian distributions by S. Blinnikov and R. Moessner ${\displaystyle y=F(x)G(x)\!}$ ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=\displaystyle \sum _{k=0}^{n}\displaystyle {\binom {n}{k}}{\dfrac {\mathrm {d} ^{n-k}}{\mathrm {d} x^{n-k}}}F(x){\dfrac {\mathrm {d} ^{k}}{\mathrm {d} x^{k}}}G(x)\!}$ ${\displaystyle y=x^{N}\!}$ ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=\displaystyle \prod _{r=1}^{n}(N-r+1)x^{N-n}\!}$ ${\displaystyle y=[F(x)]^{r}\!}$ ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=r\displaystyle {\binom {n-r}{n}}\displaystyle \sum _{j=0}^{n}{\dfrac {(-1)^{j}}{r-j}}{\displaystyle {\binom {n}{j}}[F(x)]^{r-j}{\dfrac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}[F(x)]^{j}}\!}$ ${\displaystyle y=B^{Ax}\!}$ ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}B^{Ax}\left(\ln {B}\right)^{n}\!}$ For the case of ${\displaystyle B=\exp(1)=e\!}$ (the exponential function), the above reduces to: ${\displaystyle y=e^{Ax}\!}$ ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}e^{Ax}\!}$ ${\displaystyle y=\ln[F(x)]\!}$ ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=\delta _{n}\ln[F(x)]+\displaystyle \sum _{j=1}^{n}{\dfrac {(-1)^{j-1}}{j}}{\binom {n}{j}}{\dfrac {1}{[F(x)]^{j}}}{\dfrac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}[F(x)]^{j}\!}$ where ${\displaystyle \delta _{n}={\begin{cases}1&n=0\\0&n\neq 0\\\end{cases}}\!}$ is the Kronecker delta. ${\displaystyle y=\sin(Ax+B)\!}$ ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}\sin \left(Ax+B+{\frac {n\pi }{2}}\right)\!}$ Expanding this by the sine addition formula yields a more clear form to use: ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}\left[\cos \left({\dfrac {n\pi }{2}}\right)\sin \left(Ax+B\right)+\sin \left({\dfrac {n\pi }{2}}\right)\cos \left(Ax+B\right)\right]\!}$ ${\displaystyle y=\cos(Ax+B)\!}$ ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}\cos \left(Ax+B+{\frac {n\pi }{2}}\right)\!}$ Expanding by the cosine addition formula: ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}\left[\cos \left({\dfrac {n\pi }{2}}\right)\cos \left(Ax+B\right)-\sin \left({\dfrac {n\pi }{2}}\right)\sin \left(Ax+B\right)\right]\!}$ ${\displaystyle y=\sinh(Ax+B)\!}$ ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=(-iA^{n})\sinh \left(Ax+B+{\dfrac {in\pi }{2}}\right)\!}$ ${\displaystyle y=\cosh(Ax+B)\!}$ ${\displaystyle {\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=(\pm iA^{n})\cosh \left(Ax+B\mp {\dfrac {in\pi }{2}}\right)\!}$

## Tích phân xác định

### Định nghỉa

Phép toán giải tích tìm tích phân của hàm số trong một miền xác định

${\displaystyle \int \limits _{a}^{b}f(x)dx=F=F(b)-F(a)}$

### Hoán chuyển Laplace

${\displaystyle f(t)->F(s)}$ . ${\displaystyle {\mathcal {L}}\{f(t)\}=F(s)=\int \limits _{0^{-}}^{\infty }f(t)e^{-st}dt}$
Time Domain Laplace Domain
${\displaystyle x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}}$ ${\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}}$
1 ${\displaystyle {\frac {1}{2\pi j}}\int _{\sigma -j\infty }^{\sigma +j\infty }X(s)e^{st}ds}$ ${\displaystyle \int _{-\infty }^{\infty }x(t)e^{-st}dt}$
2 ${\displaystyle \delta (t)\,}$ ${\displaystyle 1\,}$
3 ${\displaystyle \delta (t-a)\,}$ ${\displaystyle e^{-as}\,}$
4 ${\displaystyle u(t)\,}$ ${\displaystyle {\frac {1}{s}}}$
5 ${\displaystyle u(t-a)\,}$ ${\displaystyle {\frac {e^{-as}}{s}}}$
6 ${\displaystyle tu(t)\,}$ ${\displaystyle {\frac {1}{s^{2}}}}$
7 ${\displaystyle t^{n}u(t)\,}$ ${\displaystyle {\frac {n!}{s^{n+1}}}}$
8 ${\displaystyle {\frac {1}{\sqrt {\pi t}}}u(t)}$ ${\displaystyle {\frac {1}{\sqrt {s}}}}$
9 ${\displaystyle e^{at}u(t)\,}$ ${\displaystyle {\frac {1}{s-a}}}$
10 ${\displaystyle t^{n}e^{at}u(t)\,}$ ${\displaystyle {\frac {n!}{(s-a)^{n+1}}}}$
11 ${\displaystyle \cos(\omega t)u(t)\,}$ ${\displaystyle {\frac {s}{s^{2}+\omega ^{2}}}}$
12 ${\displaystyle \sin(\omega t)u(t)\,}$ ${\displaystyle {\frac {\omega }{s^{2}+\omega ^{2}}}}$
13 ${\displaystyle \cosh(\omega t)u(t)\,}$ ${\displaystyle {\frac {s}{s^{2}-\omega ^{2}}}}$
14 ${\displaystyle \sinh(\omega t)u(t)\,}$ ${\displaystyle {\frac {\omega }{s^{2}-\omega ^{2}}}}$
15 ${\displaystyle e^{at}\cos(\omega t)u(t)\,}$ ${\displaystyle {\frac {s-a}{(s-a)^{2}+\omega ^{2}}}}$
16 ${\displaystyle e^{at}\sin(\omega t)u(t)\,}$ ${\displaystyle {\frac {\omega }{(s-a)^{2}+\omega ^{2}}}}$
17 ${\displaystyle {\frac {1}{2\omega ^{3}}}(\sin \omega t-\omega t\cos \omega t)}$ ${\displaystyle {\frac {1}{(s^{2}+\omega ^{2})^{2}}}}$
18 ${\displaystyle {\frac {t}{2\omega }}\sin \omega t}$ ${\displaystyle {\frac {s}{(s^{2}+\omega ^{2})^{2}}}}$
19 ${\displaystyle {\frac {1}{2\omega }}(\sin \omega t+\omega t\cos \omega t)}$ ${\displaystyle {\frac {s^{2}}{(s^{2}+\omega ^{2})^{2}}}}$

### Hoán chuyển Fourier

${\displaystyle f(t)->F(j\omega )}$ . ${\displaystyle {\mathcal {L}}\{f(t)\}=F(s)=\int \limits _{0^{-}}^{\infty }f(t)e^{-j\omega t}dt}$

Time Domain Frequency Domain
${\displaystyle x(t)={\mathcal {F}}^{-1}\left\{X(\omega )\right\}}$ ${\displaystyle X(\omega )={\mathcal {F}}\left\{x(t)\right\}}$
1 ${\displaystyle X(j\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}dt}$ ${\displaystyle x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega }$
2 ${\displaystyle 1\,}$ ${\displaystyle 2\pi \delta (\omega )\,}$
3 ${\displaystyle -0.5+u(t)\,}$ ${\displaystyle {\frac {1}{j\omega }}\,}$
4 ${\displaystyle \delta (t)\,}$ ${\displaystyle 1\,}$
5 ${\displaystyle \delta (t-c)\,}$ ${\displaystyle e^{-j\omega c}\,}$
6 ${\displaystyle u(t)\,}$ ${\displaystyle \pi \delta (\omega )+{\frac {1}{j\omega }}\,}$
7 ${\displaystyle e^{-bt}u(t)\,(b>0)}$ ${\displaystyle {\frac {1}{j\omega +b}}\,}$
8 ${\displaystyle \cos \omega _{0}t\,}$ ${\displaystyle \pi \left[\delta (\omega +\omega _{0})+\delta (\omega -\omega _{0})\right]\,}$
9 ${\displaystyle \cos(\omega _{0}t+\theta )\,}$ ${\displaystyle \pi \left[e^{-j\theta }\delta (\omega +\omega _{0})+e^{j\theta }\delta (\omega -\omega _{0})\right]\,}$
10 ${\displaystyle \sin \omega _{0}t\,}$ ${\displaystyle j\pi \left[\delta (\omega +\omega _{0})-\delta (\omega -\omega _{0})\right]\,}$
11 ${\displaystyle \sin(\omega _{0}t+\theta )\,}$ ${\displaystyle j\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})-e^{j\theta }\delta (\omega -\omega _{0})\right]\,}$
12 ${\displaystyle {\mbox{rect}}\left({\frac {t}{\tau }}\right)\,}$ ${\displaystyle \tau {\mbox{sinc}}\left({\frac {\tau \omega }{2\pi }}\right)\,}$
13 ${\displaystyle \tau {\mbox{sinc}}\left({\frac {\tau t}{2\pi }}\right)\,}$ ${\displaystyle 2\pi {\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,}$
14 ${\displaystyle \left(1-{\frac {2|t|}{\tau }}\right){\mbox{rect}}\left({\frac {t}{\tau }}\right)\,}$ ${\displaystyle {\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau \omega }{4\pi }}\right)\,}$
15 ${\displaystyle {\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau t}{4\pi }}\right)\,}$ ${\displaystyle 2\pi \left(1-{\frac {2|\omega |}{\tau }}\right){\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,}$
16 ${\displaystyle e^{-a|t|},\Re \{a\}>0\,}$ ${\displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}\,}$
Notes:
1. ${\displaystyle {\mbox{sinc}}(x)=\sin(x)/x}$
2. ${\displaystyle {\mbox{rect}}\left({\frac {t}{\tau }}\right)}$ is the rectangular pulse function of width ${\displaystyle \tau }$
3. ${\displaystyle u(t)}$ is the Heaviside step function
4. ${\displaystyle \delta (t)}$ is the Dirac delta function

## Tích phân bất định

### Định nghỉa

Phép toán giải tích tìm tích phân của hàm số trong một miền xác định

${\displaystyle \int f(x)dx=\lim _{\Delta x\rightarrow 0}\sum [f(x)+{\frac {\Delta f(x)}{2}}]\Delta x=F(x)+C}$

### Luật toán tích phân

Table of Properties of Integrals
Rule Conditions
1 ${\displaystyle \int a\,dx=ax}$
2
Homogeniety
${\displaystyle \int af(x)\,dx=a\int f(x)\,dx}$
3
Associativity
${\displaystyle \int {\left(f\pm g\pm h\pm \cdots \right)\,dx}=\int f\,dx\pm \int g\,dx\pm \int h\,dx\pm \cdots }$
4
Integration by Parts
${\displaystyle \int _{a}^{b}fg'\,dx=\left[fg\right]_{a}^{b}-\int _{a}^{b}gf'\,dx}$
4
General Integration by Parts
${\displaystyle \int f^{(n)}g\,dx=f^{(n-1)}g'-f^{(n-2)}g''+\ldots +(-1)^{n}\int fg^{(n)}\,dx}$
5 ${\displaystyle \int f(ax)\,dx={\frac {1}{a}}\int f(x)\,dx}$
6
Substitution Rule
${\displaystyle \int g\{f(x)\}\,dx=\int g(u){\frac {dx}{du}}\,du=\int {\frac {g(u)}{f'(x)}}\,du}$ ${\displaystyle u=f(x)\,}$
7
${\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}}$ ${\displaystyle n\neq -1\,}$
8 ${\displaystyle \int {\frac {1}{x}}\,dx=\ln |x|}$
9 ${\displaystyle \int e^{x}\,dx=e^{x}}$
10 ${\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln |a|}}}$ ${\displaystyle a\neq 1}$
Notes:
1. f, g, h are functions of x
2. a, n are constants.
3. The constant of integration, C has been omitted from this table. It should be included in the working of the equation if applicable.

### Công thức tích phân bất định

#### Tích phân hàm số toán cơ bản

Integral Value Remarks
1 ${\displaystyle \int c\,dx}$ ${\displaystyle cx+C\,}$
2 ${\displaystyle \int x^{n}\,dx}$ ${\displaystyle {\frac {x^{n+1}}{n+1}}+C}$ ${\displaystyle n\neq -1}$
3 ${\displaystyle \int {\frac {1}{x}}\,dx}$ ${\displaystyle \ln {\left|x\right|}+C}$
4 ${\displaystyle \int {1 \over {a^{2}+x^{2}}}\,dx}$ ${\displaystyle {1 \over a}\arctan {x \over a}+C}$
5 ${\displaystyle \int {1 \over {\sqrt {a^{2}-x^{2}}}}\,dx}$ ${\displaystyle \arcsin {x \over a}+C}$
6 ${\displaystyle \int {-1 \over {\sqrt {a^{2}-x^{2}}}}\,dx}$ ${\displaystyle \arccos {x \over a}+C}$
7 ${\displaystyle \int {1 \over x{\sqrt {x^{2}-a^{2}}}}\,dx}$ ${\displaystyle {1 \over a}{\mbox{arcsec}}\,{|x| \over a}+C}$
8 ${\displaystyle \int \ln {x}\,dx}$ ${\displaystyle x\ln {x}-x+C\,}$
9 ${\displaystyle \int \log _{b}{x}\,dx}$ ${\displaystyle x\log _{b}{x}-x\log _{b}{e}+C\,}$
10 ${\displaystyle \int e^{x}\,dx}$ ${\displaystyle e^{x}+C\,}$
11 ${\displaystyle \int a^{x}\,dx}$ ${\displaystyle {\frac {a^{x}}{\ln {a}}}+C}$
12 ${\displaystyle \int \sin {x}\,dx}$ ${\displaystyle -\cos {x}+C\,}$
13 ${\displaystyle \int \cos {x}\,dx}$ ${\displaystyle \sin {x}+C\,}$
14 ${\displaystyle \int \tan {x}\,dx}$ ${\displaystyle -\ln {\left|\cos {x}\right|}+C\,}$
15 ${\displaystyle \int \cot {x}\,dx}$ ${\displaystyle \ln {\left|\sin {x}\right|}+C\,}$
16 ${\displaystyle \int \sec {x}\,dx}$ ${\displaystyle \ln {\left|\sec {x}+\tan {x}\right|}+C\,}$
17 ${\displaystyle \int \csc {x}\,dx}$ ${\displaystyle -\ln {\left|\csc {x}+\cot {x}\right|}+C\,}$
18 ${\displaystyle \int \sec ^{2}x\,dx}$ ${\displaystyle \tan x+C\,}$
19 ${\displaystyle \int \csc ^{2}x\,dx}$ ${\displaystyle -\cot x+C\,}$
20 ${\displaystyle \int \sec {x}\,\tan {x}\,dx}$ ${\displaystyle \sec {x}+C\,}$
21 ${\displaystyle \int \csc {x}\,\cot {x}\,dx}$ ${\displaystyle -\csc {x}+C\,}$
22 ${\displaystyle \int \sin ^{2}x\,dx}$ ${\displaystyle {\frac {1}{2}}(x-\sin x\cos x)+C\,}$
23 ${\displaystyle \int \cos ^{2}x\,dx}$ ${\displaystyle {\frac {1}{2}}(x+\sin x\cos x)+C\,}$
24 ${\displaystyle \int \sin ^{n}x\,dx}$ ${\displaystyle -{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}$
25 ${\displaystyle \int \cos ^{n}x\,dx}$ ${\displaystyle -{\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}$
26 ${\displaystyle \int \arctan {x}\,dx}$ ${\displaystyle x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}$
27 ${\displaystyle \int \sinh x\,dx}$ ${\displaystyle \cosh x+C\,}$
28 ${\displaystyle \int \cosh x\,dx}$ ${\displaystyle \sinh x+C\,}$
29 ${\displaystyle \int \tanh x\,dx}$ ${\displaystyle \ln |\cosh x|+C\,}$
30 ${\displaystyle \int {\mbox{csch}}\,x\,dx}$ ${\displaystyle \ln \left|\tanh {x \over 2}\right|+C}$
31 ${\displaystyle \int {\mbox{sech}}\,x\,dx}$ ${\displaystyle \arctan(\sinh x)+C\,}$
32 ${\displaystyle \int \coth x\,dx}$ ${\displaystyle \ln |\sinh x|+C\,}$

#### Tích phân hàm số toán sine

${\displaystyle \int \sin ax\;dx=-{\frac {1}{a}}\cos ax+C\,\!}$
${\displaystyle \int \sin ^{2}{ax}\;dx={\frac {x}{2}}-{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}-{\frac {1}{2a}}\sin ax\cos ax+C\!}$
${\displaystyle \int x\sin ^{2}{ax}\;dx={\frac {x^{2}}{4}}-{\frac {x}{4a}}\sin 2ax-{\frac {1}{8a^{2}}}\cos 2ax+C\!}$
${\displaystyle \int x^{2}\sin ^{2}{ax}\;dx={\frac {x^{3}}{6}}-\left({\frac {x^{2}}{4a}}-{\frac {1}{8a^{3}}}\right)\sin 2ax-{\frac {x}{4a^{2}}}\cos 2ax+C\!}$
${\displaystyle \int \sin b_{1}x\sin b_{2}x\;dx={\frac {\sin((b_{1}-b_{2})x)}{2(b_{1}-b_{2})}}-{\frac {\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}}+C\qquad {\mbox{(for }}|b_{1}|\neq |b_{2}|{\mbox{)}}\,\!}$
${\displaystyle \int \sin ^{n}{ax}\;dx=-{\frac {\sin ^{n-1}ax\cos ax}{na}}+{\frac {n-1}{n}}\int \sin ^{n-2}ax\;dx\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {dx}{\sin ax}}={\frac {1}{a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}$
${\displaystyle \int {\frac {dx}{\sin ^{n}ax}}={\frac {\cos ax}{a(1-n)\sin ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}ax}}\qquad {\mbox{(for }}n>1{\mbox{)}}\,\!}$
${\displaystyle \int x\sin ax\;dx={\frac {\sin ax}{a^{2}}}-{\frac {x\cos ax}{a}}+C\,\!}$
${\displaystyle \int x^{n}\sin ax\;dx=-{\frac {x^{n}}{a}}\cos ax+{\frac {n}{a}}\int x^{n-1}\cos ax\;dx\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}$
${\displaystyle \int _{\frac {-a}{2}}^{\frac {a}{2}}x^{2}\sin ^{2}{\frac {n\pi x}{a}}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(for }}n=2,4,6...{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {\sin ax}{x}}dx=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!}}+C\,\!}$
${\displaystyle \int {\frac {\sin ax}{x^{n}}}dx=-{\frac {\sin ax}{(n-1)x^{n-1}}}+{\frac {a}{n-1}}\int {\frac {\cos ax}{x^{n-1}}}dx\,\!}$
${\displaystyle \int {\frac {dx}{1\pm \sin ax}}={\frac {1}{a}}\tan \left({\frac {ax}{2}}\mp {\frac {\pi }{4}}\right)+C}$
${\displaystyle \int {\frac {x\;dx}{1+\sin ax}}={\frac {x}{a}}\tan \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{a^{2}}}\ln \left|\cos \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)\right|+C}$
${\displaystyle \int {\frac {x\;dx}{1-\sin ax}}={\frac {x}{a}}\cot \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)+{\frac {2}{a^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)\right|+C}$
${\displaystyle \int {\frac {\sin ax\;dx}{1\pm \sin ax}}=\pm x+{\frac {1}{a}}\tan \left({\frac {\pi }{4}}\mp {\frac {ax}{2}}\right)+C}$

${\displaystyle \int \cos ax\;dx={\frac {1}{a}}\sin ax+C\,\!}$
##### Tích phân hàm số toán cosine
${\displaystyle \int \cos ^{n}ax\;dx={\frac {\cos ^{n-1}ax\sin ax}{na}}+{\frac {n-1}{n}}\int \cos ^{n-2}ax\;dx\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}$
${\displaystyle \int x\cos ax\;dx={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+C\,\!}$
${\displaystyle \int \cos ^{2}{ax}\;dx={\frac {x}{2}}+{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}+{\frac {1}{2a}}\sin ax\cos ax+C\!}$
${\displaystyle \int x^{2}\cos ^{2}{ax}\;dx={\frac {x^{3}}{6}}+\left({\frac {x^{2}}{4a}}-{\frac {1}{8a^{3}}}\right)\sin 2ax+{\frac {x}{4a^{2}}}\cos 2ax+C\!}$
${\displaystyle \int x^{n}\cos ax\;dx={\frac {x^{n}\sin ax}{a}}-{\frac {n}{a}}\int x^{n-1}\sin ax\;dx\,\!}$
${\displaystyle \int {\frac {\cos ax}{x}}dx=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}}{2k\cdot (2k)!}}+C\,\!}$
${\displaystyle \int {\frac {\cos ax}{x^{n}}}dx=-{\frac {\cos ax}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\int {\frac {\sin ax}{x^{n-1}}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {dx}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}$
${\displaystyle \int {\frac {dx}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cos ^{n-2}ax}}\qquad {\mbox{(for }}n>1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {dx}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}$
${\displaystyle \int {\frac {dx}{1-\cos ax}}=-{\frac {1}{a}}\cot {\frac {ax}{2}}+C\,\!}$
${\displaystyle \int {\frac {x\;dx}{1+\cos ax}}={\frac {x}{a}}\tan {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\cos {\frac {ax}{2}}\right|+C}$
${\displaystyle \int {\frac {x\;dx}{1-\cos ax}}=-{\frac {x}{a}}\cot {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\sin {\frac {ax}{2}}\right|+C}$
${\displaystyle \int {\frac {\cos ax\;dx}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}$
${\displaystyle \int {\frac {\cos ax\;dx}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+C\,\!}$
${\displaystyle \int \cos a_{1}x\cos a_{2}x\;dx={\frac {\sin(a_{1}-a_{2})x}{2(a_{1}-a_{2})}}+{\frac {\sin(a_{1}+a_{2})x}{2(a_{1}+a_{2})}}+C\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}$
##### Tích phân hàm số toán tangent
${\displaystyle \int \tan ax\;dx=-{\frac {1}{a}}\ln |\cos ax|+C={\frac {1}{a}}\ln |\sec ax|+C\,\!}$
${\displaystyle \int \tan ^{n}ax\;dx={\frac {1}{a(n-1)}}\tan ^{n-1}ax-\int \tan ^{n-2}ax\;dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {dx}{q\tan ax+p}}={\frac {1}{p^{2}+q^{2}}}(px+{\frac {q}{a}}\ln |q\sin ax+p\cos ax|)+C\qquad {\mbox{(for }}p^{2}+q^{2}\neq 0{\mbox{)}}\,\!}$

${\displaystyle \int {\frac {dx}{\tan ax}}={\frac {1}{a}}\ln |\sin ax|+C\,\!}$
${\displaystyle \int {\frac {dx}{\tan ax+1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}$
${\displaystyle \int {\frac {dx}{\tan ax-1}}=-{\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}$
${\displaystyle \int {\frac {\tan ax\;dx}{\tan ax+1}}={\frac {x}{2}}-{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}$
${\displaystyle \int {\frac {\tan ax\;dx}{\tan ax-1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}$
##### Tích phân hàm số toán secant
${\displaystyle \int \sec {ax}\,dx={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C}$
${\displaystyle \int \sec ^{n}{ax}\,dx={\frac {\sec ^{n-1}{ax}\sin {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{ax}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}$
${\displaystyle \int \sec ^{n}{x}\,dx={\frac {\sec ^{n-2}{x}\tan {x}}{n-1}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{x}\,dx}$
${\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C}$
##### Tích phân hàm số toán cosecant
${\displaystyle \int \csc {ax}\,dx={\frac {1}{a}}\ln {\left|\csc {ax}-\cot {ax}\right|}+C}$
${\displaystyle \int \csc ^{2}{x}\,dx=-\cot {x}+C}$
${\displaystyle \int \csc ^{n}{ax}\,dx=-{\frac {\csc ^{n-1}{ax}\cos {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{ax}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}$
##### Tích phân hàm số toán cotangent
${\displaystyle \int \cot ax\;dx={\frac {1}{a}}\ln |\sin ax|+C\,\!}$
${\displaystyle \int \cot ^{n}ax\;dx=-{\frac {1}{a(n-1)}}\cot ^{n-1}ax-\int \cot ^{n-2}ax\;dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {dx}{1+\cot ax}}=\int {\frac {\tan ax\;dx}{\tan ax+1}}\,\!}$
${\displaystyle \int {\frac {dx}{1-\cot ax}}=\int {\frac {\tan ax\;dx}{\tan ax-1}}\,\!}$

${\displaystyle \int {\frac {dx}{\cos ax\pm \sin ax}}={\frac {1}{a{\sqrt {2}}}}\ln \left|\tan \left({\frac {ax}{2}}\pm {\frac {\pi }{8}}\right)\right|+C}$
${\displaystyle \int {\frac {dx}{(\cos ax\pm \sin ax)^{2}}}={\frac {1}{2a}}\tan \left(ax\mp {\frac {\pi }{4}}\right)+C}$
${\displaystyle \int {\frac {dx}{(\cos x+\sin x)^{n}}}={\frac {1}{n-1}}\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1}}}-2(n-2)\int {\frac {dx}{(\cos x+\sin x)^{n-2}}}\right)}$
${\displaystyle \int {\frac {\cos ax\;dx}{\cos ax+\sin ax}}={\frac {x}{2}}+{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}$
${\displaystyle \int {\frac {\cos ax\;dx}{\cos ax-\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}$
${\displaystyle \int {\frac {\sin ax\;dx}{\cos ax+\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}$
${\displaystyle \int {\frac {\sin ax\;dx}{\cos ax-\sin ax}}=-{\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}$
${\displaystyle \int {\frac {\cos ax\;dx}{\sin ax(1+\cos ax)}}=-{\frac {1}{4a}}\tan ^{2}{\frac {ax}{2}}+{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}$
${\displaystyle \int {\frac {\cos ax\;dx}{\sin ax(1+-\cos ax)}}=-{\frac {1}{4a}}\cot ^{2}{\frac {ax}{2}}-{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}$
${\displaystyle \int {\frac {\sin ax\;dx}{\cos ax(1+\sin ax)}}={\frac {1}{4a}}\cot ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}$
${\displaystyle \int {\frac {\sin ax\;dx}{\cos ax(1-\sin ax)}}={\frac {1}{4a}}\tan ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}$
${\displaystyle \int \sin ax\cos ax\;dx={\frac {1}{2a}}\sin ^{2}ax+C\,\!}$
${\displaystyle \int \sin a_{1}x\cos a_{2}x\;dx=-{\frac {\cos(a_{1}+a_{2})x}{2(a_{1}+a_{2})}}-{\frac {\cos(a_{1}-a_{2})x}{2(a_{1}-a_{2})}}+C\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}$
${\displaystyle \int \sin ^{n}ax\cos ax\;dx={\frac {1}{a(n+1)}}\sin ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}$
${\displaystyle \int \sin ax\cos ^{n}ax\;dx=-{\frac {1}{a(n+1)}}\cos ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}$
${\displaystyle \int \sin ^{n}ax\cos ^{m}ax\;dx=-{\frac {\sin ^{n-1}ax\cos ^{m+1}ax}{a(n+m)}}+{\frac {n-1}{n+m}}\int \sin ^{n-2}ax\cos ^{m}ax\;dx\qquad {\mbox{(for }}m,n>0{\mbox{)}}\,\!}$
also: ${\displaystyle \int \sin ^{n}ax\cos ^{m}ax\;dx={\frac {\sin ^{n+1}ax\cos ^{m-1}ax}{a(n+m)}}+{\frac {m-1}{n+m}}\int \sin ^{n}ax\cos ^{m-2}ax\;dx\qquad {\mbox{(for }}m,n>0{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {dx}{\sin ax\cos ax}}={\frac {1}{a}}\ln \left|\tan ax\right|+C}$
${\displaystyle \int {\frac {dx}{\sin ax\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+\int {\frac {dx}{\sin ax\cos ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {dx}{\sin ^{n}ax\cos ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+\int {\frac {dx}{\sin ^{n-2}ax\cos ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {\sin ax\;dx}{\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {\sin ^{2}ax\;dx}{\cos ax}}=-{\frac {1}{a}}\sin ax+{\frac {1}{a}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {ax}{2}}\right)\right|+C}$
${\displaystyle \int {\frac {\sin ^{2}ax\;dx}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}-{\frac {1}{n-1}}\int {\frac {dx}{\cos ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ax}}=-{\frac {\sin ^{n-1}ax}{a(n-1)}}+\int {\frac {\sin ^{n-2}ax\;dx}{\cos ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ^{m}ax}}={\frac {\sin ^{n+1}ax}{a(m-1)\cos ^{m-1}ax}}-{\frac {n-m+2}{m-1}}\int {\frac {\sin ^{n}ax\;dx}{\cos ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}$
also: ${\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ^{m}ax}}=-{\frac {\sin ^{n-1}ax}{a(n-m)\cos ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}ax\;dx}{\cos ^{m}ax}}\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,\!}$
also: ${\displaystyle \int {\frac {\sin ^{n}ax\;dx}{\cos ^{m}ax}}={\frac {\sin ^{n-1}ax}{a(m-1)\cos ^{m-1}ax}}-{\frac {n-1}{m-1}}\int {\frac {\sin ^{n-2}ax\;dx}{\cos ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {\cos ax\;dx}{\sin ^{n}ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {\cos ^{2}ax\;dx}{\sin ax}}={\frac {1}{a}}\left(\cos ax+\ln \left|\tan {\frac {ax}{2}}\right|\right)+C}$
${\displaystyle \int {\frac {\cos ^{2}ax\;dx}{\sin ^{n}ax}}=-{\frac {1}{n-1}}\left({\frac {\cos ax}{a\sin ^{n-1}ax)}}+\int {\frac {dx}{\sin ^{n-2}ax}}\right)\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}$
${\displaystyle \int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}}=-{\frac {\cos ^{n+1}ax}{a(m-1)\sin ^{m-1}ax}}-{\frac {n-m-2}{m-1}}\int {\frac {\cos ^{n}ax\;dx}{\sin ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}$
also: ${\displaystyle \int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}}={\frac {\cos ^{n-1}ax}{a(n-m)\sin ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cos ^{n-2}ax\;dx}{\sin ^{m}ax}}\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,\!}$
also: ${\displaystyle \int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}}=-{\frac {\cos ^{n-1}ax}{a(m-1)\sin ^{m-1}ax}}-{\frac {n-1}{m-1}}\int {\frac {\cos ^{n-2}ax\;dx}{\sin ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}$

${\displaystyle \int \sin ax\tan ax\;dx={\frac {1}{a}}(\ln |\sec ax+\tan ax|-\sin ax)+C\,\!}$
${\displaystyle \int {\frac {\tan ^{n}ax\;dx}{\sin ^{2}ax}}={\frac {1}{a(n-1)}}\tan ^{n-1}(ax)+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}$

${\displaystyle \int {\frac {\tan ^{n}ax\;dx}{\cos ^{2}ax}}={\frac {1}{a(n+1)}}\tan ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}$

${\displaystyle \int {\frac {\cot ^{n}ax\;dx}{\sin ^{2}ax}}={\frac {1}{a(n+1)}}\cot ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}$

${\displaystyle \int {\frac {\cot ^{n}ax\;dx}{\cos ^{2}ax}}={\frac {1}{a(1-n)}}\tan ^{1-n}ax+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}$

${\displaystyle \int _{-c}^{c}\sin {x}\;dx=0\!}$
${\displaystyle \int _{-c}^{c}\cos {x}\;dx=2\int _{0}^{c}\cos {x}\;dx=2\int _{-c}^{0}\cos {x}\;dx=2\sin {c}\!}$
${\displaystyle \int _{-c}^{c}\tan {x}\;dx=0\!}$
${\displaystyle \int _{\frac {-a}{2}}^{\frac {a}{2}}x^{2}\cos ^{2}{\frac {n\pi x}{a}}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(for }}n=1,3,5...{\mbox{)}}\,\!}$

${\displaystyle \int \arcsin {\frac {x}{c}}\,dx=x\arcsin {\frac {x}{c}}+{\sqrt {c^{2}-x^{2}}}}$
${\displaystyle \int x\arcsin {\frac {x}{c}}\,dx=\left({\frac {x^{2}}{2}}-{\frac {c^{2}}{4}}\right)\arcsin {\frac {x}{c}}+{\frac {x}{4}}{\sqrt {c^{2}-x^{2}}}}$
${\displaystyle \int x^{2}\arcsin {\frac {x}{c}}\,dx={\frac {x^{3}}{3}}\arcsin {\frac {x}{c}}+{\frac {x^{2}+2c^{2}}{9}}{\sqrt {c^{2}-x^{2}}}}$
${\displaystyle \int x^{n}\sin ^{-1}x\,dx={\frac {1}{n+1}}\left(x^{n+1}\sin ^{-1}x\right.}$
${\displaystyle \left.+{\frac {x^{n}{\sqrt {1-x^{2}}}-nx^{n-1}\sin ^{-1}x}{n-1}}+n\int x^{n-2}\sin ^{-1}x\,dx\right)}$
${\displaystyle \int \arccos {\frac {x}{c}}\,dx=x\arccos {\frac {x}{c}}-{\sqrt {c^{2}-x^{2}}}}$
${\displaystyle \int x\arccos {\frac {x}{c}}\,dx=\left({\frac {x^{2}}{2}}-{\frac {c^{2}}{4}}\right)\arccos {\frac {x}{c}}-{\frac {x}{4}}{\sqrt {c^{2}-x^{2}}}}$
${\displaystyle \int x^{2}\arccos {\frac {x}{c}}\,dx={\frac {x^{3}}{3}}\arccos {\frac {x}{c}}-{\frac {x^{2}+2c^{2}}{9}}{\sqrt {c^{2}-x^{2}}}}$
${\displaystyle \int \arctan {\frac {x}{c}}\,dx=x\arctan {\frac {x}{c}}-{\frac {c}{2}}\ln(c^{2}+x^{2})}$
${\displaystyle \int x\arctan {\frac {x}{c}}\,dx={\frac {c^{2}+x^{2}}{2}}\arctan {\frac {x}{c}}-{\frac {cx}{2}}}$
${\displaystyle \int x^{2}\arctan {\frac {x}{c}}\,dx={\frac {x^{3}}{3}}\arctan {\frac {x}{c}}-{\frac {cx^{2}}{6}}+{\frac {c^{3}}{6}}\ln {c^{2}+x^{2}}}$
${\displaystyle \int x^{n}\arctan {\frac {x}{c}}\,dx={\frac {x^{n+1}}{n+1}}\arctan {\frac {x}{c}}-{\frac {c}{n+1}}\int {\frac {x^{n+1}dx}{c^{2}+x^{2}}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$
${\displaystyle \int \operatorname {arcsec} {\frac {x}{c}}\,dx=x\operatorname {arcsec} {\frac {x}{c}}+{\frac {x}{c|x|}}\ln {|x\pm {\sqrt {x^{2}-1}}|}}$
${\displaystyle \int x\operatorname {arcsec} {x}\,dx\,=\,{\frac {1}{2}}\left(x^{2}\operatorname {arcsec} {x}-{\sqrt {x^{2}-1}}\right)}$
${\displaystyle \int x^{n}\operatorname {arcsec} {x}\,dx\,=\,{\frac {1}{n+1}}\left(x^{n+1}\operatorname {arcsec} {x}-{\frac {1}{n}}\left(x^{n-1}{\sqrt {x^{2}-1}}\;\right.\right.}$
${\displaystyle \left.\left.+(1-n)\left(x^{n-1}\operatorname {arcsec} {x}+(1-n)\int x^{n-2}\operatorname {arcsec} {x}\,dx\right)\right)\right)}$
${\displaystyle \int \mathrm {arccot} \,{\frac {x}{c}}\,dx=x\,\mathrm {arccot} \,{\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})}$
${\displaystyle \int x\,\mathrm {arccot} \,{\frac {x}{c}}\,dx={\frac {c^{2}+x^{2}}{2}}\,\mathrm {arccot} \,{\frac {x}{c}}+{\frac {cx}{2}}}$
${\displaystyle \int x^{2}\,\mathrm {arccot} \,{\frac {x}{c}}\,dx={\frac {x^{3}}{3}}\,\mathrm {arccot} \,{\frac {x}{c}}+{\frac {cx^{2}}{6}}-{\frac {c^{3}}{6}}\ln(c^{2}+x^{2})}$
${\displaystyle \int x^{n}\,\mathrm {arccot} \,{\frac {x}{c}}\,dx={\frac {x^{n+1}}{n+1}}\,\mathrm {arccot} \,{\frac {x}{c}}+{\frac {c}{n+1}}\int {\frac {x^{n+1}dx}{c^{2}+x^{2}}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int (ax+b)^{n}dx={\frac {(ax+b)^{n+1}}{a(n+1)}}\qquad {\mbox{( }}n\neq -1{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {dx}{ax+b}}={\frac {1}{a}}\ln \left|ax+b\right|}$
${\displaystyle \int x(ax+b)^{n}dx={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}\qquad {\mbox{( }}n\not \in \{1,2\}{\mbox{)}}}$
${\displaystyle \int {\frac {x}{ax+b}}dx={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|}$
${\displaystyle \int {\frac {x}{(ax+b)^{2}}}dx={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln \left|ax+b\right|}$
${\displaystyle \int {\frac {x}{(ax+b)^{n}}}dx={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}\qquad {\mbox{( }}n\not \in \{1,2\}{\mbox{)}}}$
${\displaystyle \int {\frac {x^{2}}{ax+b}}dx={\frac {1}{a^{3}}}\left({\frac {(ax+b)^{2}}{2}}-2b(ax+b)+b^{2}\ln \left|ax+b\right|\right)}$
${\displaystyle \int {\frac {x^{2}}{(ax+b)^{2}}}dx={\frac {1}{a^{3}}}\left(ax+b-2b\ln \left|ax+b\right|-{\frac {b^{2}}{ax+b}}\right)}$
${\displaystyle \int {\frac {x^{2}}{(ax+b)^{3}}}dx={\frac {1}{a^{3}}}\left(\ln \left|ax+b\right|+{\frac {2b}{ax+b}}-{\frac {b^{2}}{2(ax+b)^{2}}}\right)}$
${\displaystyle \int {\frac {x^{2}}{(ax+b)^{n}}}dx={\frac {1}{a^{3}}}\left(-{\frac {1}{(n-3)(ax+b)^{n-3}}}+{\frac {2b}{(n-2)(a+b)^{n-2}}}-{\frac {b^{2}}{(n-1)(ax+b)^{n-1}}}\right)\qquad {\mbox{( }}n\not \in \{1,2,3\}{\mbox{)}}}$
${\displaystyle \int {\frac {dx}{x(ax+b)}}=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|}$
${\displaystyle \int {\frac {dx}{x^{2}(ax+b)}}=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|}$
${\displaystyle \int {\frac {dx}{x^{2}(ax+b)^{2}}}=-a\left({\frac {1}{b^{2}(ax+b)}}+{\frac {1}{ab^{2}x}}-{\frac {2}{b^{3}}}\ln \left|{\frac {ax+b}{x}}\right|\right)}$
${\displaystyle \int {\frac {dx}{x^{2}+a^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}\,\!}$
${\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {arctanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}\qquad {\mbox{( }}|x|<|a|{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\,\mathrm {arccoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}\qquad {\mbox{( }}|x|>|a|{\mbox{)}}\,\!}$
${\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{( }}4ac-b^{2}>0{\mbox{)}}}$
${\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {b^{2}-4ac}}}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}={\frac {1}{\sqrt {b^{2}-4ac}}}\ln \left|{\frac {2ax+b-{\sqrt {b^{2}-4ac}}}{2ax+b+{\sqrt {b^{2}-4ac}}}}\right|\qquad {\mbox{( }}4ac-b^{2}<0{\mbox{)}}}$
${\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}=-{\frac {2}{2ax+b}}\qquad {\mbox{( }}4ac-b^{2}=0{\mbox{)}}}$
${\displaystyle \int {\frac {x}{ax^{2}+bx+c}}dx={\frac {1}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {b}{2a}}\int {\frac {dx}{ax^{2}+bx+c}}}$
${\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {4ac-b^{2}}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{( }}4ac-b^{2}>0{\mbox{)}}}$
${\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}}}}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{( }}4ac-b^{2}<0{\mbox{)}}}$
${\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a(2ax+b)}}\qquad {\mbox{( }}4ac-b^{2}=0{\mbox{)}}}$
${\displaystyle \int {\frac {dx}{(ax^{2}+bx+c)^{n}}}={\frac {2ax+b}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}+{\frac {(2n-3)2a}{(n-1)(4ac-b^{2})}}\int {\frac {dx}{(ax^{2}+bx+c)^{n-1}}}\,\!}$
${\displaystyle \int {\frac {x}{(ax^{2}+bx+c)^{n}}}dx={\frac {bx+2c}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}-{\frac {b(2n-3)}{(n-1)(4ac-b^{2})}}\int {\frac {dx}{(ax^{2}+bx+c)^{n-1}}}\,\!}$
${\displaystyle \int {\frac {dx}{x(ax^{2}+bx+c)}}={\frac {1}{2c}}\ln \left|{\frac {x^{2}}{ax^{2}+bx+c}}\right|-{\frac {b}{2c}}\int {\frac {dx}{ax^{2}+bx+c}}}$

${\displaystyle \int r\;dx={\frac {1}{2}}\left(xr+a^{2}\,\ln \left(x+r\right)\right)}$
${\displaystyle \int r^{3}\;dx={\frac {1}{4}}xr^{3}+{\frac {3}{8}}a^{2}xr+{\frac {3}{8}}a^{4}\ln \left(x+r\right)}$
${\displaystyle \int r^{5}\;dx={\frac {1}{6}}xr^{5}+{\frac {5}{24}}a^{2}xr^{3}+{\frac {5}{16}}a^{4}xr+{\frac {5}{16}}a^{6}\ln \left(x+r\right)}$
${\displaystyle \int xr\;dx={\frac {r^{3}}{3}}}$
${\displaystyle \int xr^{3}\;dx={\frac {r^{5}}{5}}}$
${\displaystyle \int xr^{2n+1}\;dx={\frac {r^{2n+3}}{2n+3}}}$
${\displaystyle \int x^{2}r\;dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left(x+r\right)}$
${\displaystyle \int x^{2}r^{3}\;dx={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left(x+r\right)}$
${\displaystyle \int x^{3}r\;dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}}$
${\displaystyle \int x^{3}r^{3}\;dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}}$
${\displaystyle \int x^{3}r^{2n+1}\;dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{3}r^{2n+3}}{2n+3}}}$
${\displaystyle \int x^{4}r\;dx={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}}+{\frac {a^{4}xr}{16}}+{\frac {a^{6}}{16}}\ln \left(x+r\right)}$
${\displaystyle \int x^{4}r^{3}\;dx={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}}+{\frac {a^{4}xr^{3}}{64}}+{\frac {3a^{6}xr}{128}}+{\frac {3a^{8}}{128}}\ln \left(x+r\right)}$
${\displaystyle \int x^{5}r\;dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}}$
${\displaystyle \int x^{5}r^{3}\;dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}}$
${\displaystyle \int x^{5}r^{2n+1}\;dx={\frac {r^{2n+7}}{2n+7}}-{\frac {2a^{2}r^{2n+5}}{2n+5}}+{\frac {a^{4}r^{2n+3}}{2n+3}}}$
${\displaystyle \int {\frac {r\;dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\,\operatorname {arsinh} {\frac {a}{x}}}$
${\displaystyle \int {\frac {r^{3}\;dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|}$
${\displaystyle \int {\frac {r^{5}\;dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{2}r^{3}}{3}}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|}$
${\displaystyle \int {\frac {r^{7}\;dx}{x}}={\frac {r^{7}}{7}}+{\frac {a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|}$
${\displaystyle \int {\frac {dx}{r}}=\operatorname {arsinh} {\frac {x}{a}}=\ln \left({\frac {x+r}{a}}\right)}$
${\displaystyle \int {\frac {dx}{r^{3}}}={\frac {x}{a^{2}r}}}$
${\displaystyle \int {\frac {x\,dx}{r}}=r}$
${\displaystyle \int {\frac {x\,dx}{r^{3}}}=-{\frac {1}{r}}}$
${\displaystyle \int {\frac {x^{2}\;dx}{r}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\,\operatorname {arsinh} {\frac {x}{a}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left({\frac {x+r}{a}}\right)}$
${\displaystyle \int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\operatorname {arsinh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|}$
##### Tích phân hàm số toán nghịch
${\displaystyle \int \mathrm {arsinh} \,{\frac {x}{c}}\,dx=x\,\mathrm {arsinh} \,{\frac {x}{c}}-{\sqrt {x^{2}+c^{2}}}}$
${\displaystyle \int \mathrm {arcosh} \,{\frac {x}{c}}\,dx=x\,\mathrm {arcosh} \,{\frac {x}{c}}-{\sqrt {x^{2}-c^{2}}}}$
${\displaystyle \int \mathrm {artanh} \,{\frac {x}{c}}\,dx=x\,\mathrm {artanh} \,{\frac {x}{c}}+{\frac {c}{2}}\ln |c^{2}-x^{2}|\qquad {\mbox{( }}|x|<|c|{\mbox{)}}}$
${\displaystyle \int \mathrm {arcoth} \,{\frac {x}{c}}\,dx=x\,\mathrm {arcoth} \,{\frac {x}{c}}+{\frac {c}{2}}\ln |x^{2}-c^{2}|\qquad {\mbox{( }}|x|>|c|{\mbox{)}}}$
${\displaystyle \int \mathrm {arsech} \,{\frac {x}{c}}\,dx=x\,\mathrm {arsech} \,{\frac {x}{c}}-c\,\mathrm {arctan} \,{\frac {x\,{\sqrt {\frac {c-x}{c+x}}}}{x-c}}\qquad {\mbox{( }}x\in (0,\,c){\mbox{)}}}$
${\displaystyle \int \mathrm {arcsch} \,{\frac {x}{c}}\,dx=x\,\mathrm {arcsch} \,{\frac {x}{c}}+c\,\ln \,{\frac {x+{\sqrt {x^{2}+c^{2}}}}{c}}\qquad {\mbox{( }}x\in (0,\,c){\mbox{)}}}$
##### Tích phân hàm số toán Ln

Chú ý: bài này quy ước x>0.

${\displaystyle \int \ln cx\,dx=x\ln cx-x}$
• ${\displaystyle \int (\ln x)^{2}\;dx=x(\ln x)^{2}-2x\ln x+2x}$
• ${\displaystyle \int (\ln cx)^{n}\;dx=x(\ln cx)^{n}-n\int (\ln cx)^{n-1}dx}$
• ${\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{i=2}^{\infty }{\frac {(\ln x)^{i}}{i\cdot i!}}}$
• ${\displaystyle \int {\frac {dx}{(\ln x)^{n}}}=-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {dx}{(\ln x)^{n-1}}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$
• ${\displaystyle \int x^{m}\ln x\;dx=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)\qquad {\mbox{( }}m\neq -1{\mbox{)}}}$
• ${\displaystyle \int x^{m}(\ln x)^{n}\;dx={\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}dx\qquad {\mbox{( }}m\neq -1{\mbox{)}}}$
• ${\displaystyle \int {\frac {(\ln x)^{n}\;dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{( }}n\neq -1{\mbox{)}}}$
• ${\displaystyle \int {\frac {\ln x\,dx}{x^{m}}}=-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}}$
• ${\displaystyle \int {\frac {(\ln x)^{n}\;dx}{x^{m}}}=-{\frac {(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\ln x)^{n-1}dx}{x^{m}}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}}$
• ${\displaystyle \int {\frac {x^{m}\;dx}{(\ln x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}dx}{(\ln x)^{n-1}}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$
• ${\displaystyle \int {\frac {dx}{x\ln x}}=\ln |\ln x|}$
• ${\displaystyle \int {\frac {dx}{x^{n}\ln x}}=\ln |\ln x|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(n-1)^{i}(\ln x)^{i}}{i\cdot i!}}}$
• ${\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$
• ${\displaystyle \int \sin(\ln x)\;dx={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))}$
• ${\displaystyle \int \cos(\ln x)\;dx={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}$
##### Tích phân hàm số toán lủy thừa e
${\displaystyle \int e^{cx}\;dx={\frac {1}{c}}e^{cx}}$
${\displaystyle \int a^{cx}\;dx={\frac {1}{c\ln a}}a^{cx}\qquad {\mbox{( }}a>0,{\mbox{ }}a\neq 1{\mbox{)}}}$
${\displaystyle \int xe^{cx}\;dx={\frac {e^{cx}}{c^{2}}}(cx-1)}$
${\displaystyle \int x^{2}e^{cx}\;dx=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)}$
${\displaystyle \int x^{n}e^{cx}\;dx={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}dx}$
${\displaystyle \int {\frac {e^{cx}\;dx}{x}}=\ln |x|+\sum _{i=1}^{\infty }{\frac {(cx)^{i}}{i\cdot i!}}}$
${\displaystyle \int {\frac {e^{cx}\;dx}{x^{n}}}={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}}{x^{n-1}}}\,dx\right)\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$
${\displaystyle \int e^{cx}\ln x\;dx={\frac {1}{c}}e^{cx}\ln |x|-\operatorname {Ei} \,(cx)}$
${\displaystyle \int e^{cx}\sin bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)}$
${\displaystyle \int e^{cx}\cos bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)}$
${\displaystyle \int e^{cx}\sin ^{n}x\;dx={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\;dx}$
${\displaystyle \int e^{cx}\cos ^{n}x\;dx={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\;dx}$
${\displaystyle \int xe^{cx^{2}}\;dx={\frac {1}{2c}}\;e^{cx^{2}}}$
${\displaystyle \int {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\;dx={\frac {1}{2\sigma }}(1+{\mbox{erf}}\,{\frac {x-\mu }{\sigma {\sqrt {2}}}})}$