Thảo luận:Nhập môn Giải tích/Tích phân/Tích phân bất định/Tích phân hàm số cơ bản

Integrals of simple functions

C is used for an arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives.

These formulas only state in another form the assertions in the table of derivatives.

Rational functions

more integrals: List of integrals of rational functions

$\int a\,dx=ax+C$

$\int x^{a}\,dx={\frac {x^{a+1}}{a+1}}+C$

$\int {1 \over x}\,dx=\ln \left|x\right|+C$

Exponential functions

more integrals: List of integrals of exponential functions

$\int e^{x}\,dx=e^{x}+C$

$\int a^{x}\,dx={\frac {a^{x}}{\ln a}}+C$

Logarithms

more integrals: List of integrals of logarithmic functions

$\int \ln x\,dx=x|lnx-x|+C$

$\int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C$

Trigonometric functions

more integrals: List of integrals of trigonometric functions

$\int \sin {x}\,dx=-\cos {x}+C$

$\int \cos {x}\,dx=\sin {x}+C$

$\int \tan {x}\,dx=-\ln {\left|\cos {x}\right|}+C=\ln {\left|\sec {x}\right|}+C$

$\int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C$

$\int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C$

$\int \csc {x}\,dx=-\ln {\left|\csc {x}+\cot {x}\right|}+C$

$\int \sec ^{2}x\,dx=\tan x+C$

$\int \csc ^{2}x\,dx=-\cot x+C$

$\int \sec {x}\,\tan {x}\,dx=\sec {x}+C$

$\int \csc {x}\,\cot {x}\,dx=-\csc {x}+C$

$\int \sin ^{2}x\,dx={\frac {1}{2}}\left(x-{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x-\sin x\cos x)+C$

$\int \cos ^{2}x\,dx={\frac {1}{2}}\left(x+{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x+\sin x\cos x)+C$

$\int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C$

(see integral of secant cubed)

$\int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx$

$\int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx$

Inverse trigonometric functions

more integrals: List of integrals of inverse trigonometric functions

$\int \arcsin {x}\,dx=x\,\arcsin {x}+{\sqrt {1-x^{2}}}+C$

$\int \arccos {x}\,dx=x\,\arccos {x}-{\sqrt {1-x^{2}}}+C$

$\int \arctan {x}\,dx=x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C$

$\int \operatorname {arcsec} {x}\,dx=x\,\operatorname {arcsec} {x}+{\frac {{\sqrt {x^{2}-1}}\ln {(x+{\sqrt {x^{2}-1}})}}{x\,{\sqrt {1-{\frac {1}{x^{2}}}}}}}+C$

$\int \operatorname {arccsc} {x}\,dx=x\,\operatorname {arccsc} {x}+{\frac {{\sqrt {x^{2}-1}}\ln {(x+{\sqrt {x^{2}-1}})}}{x\,{\sqrt {1-{\frac {1}{x^{2}}}}}}}+C$

$\int \operatorname {arccot} {x}\,dx=x\,\operatorname {arccot} {x}+{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C$

Hyperbolic functions

more integrals: List of integrals of hyperbolic functions

$\int \sinh x\,dx=\cosh x+C$

$\int \cosh x\,dx=\sinh x+C$

$\int \tanh x\,dx=\ln |\cosh x|+C$

$\int {\mbox{cosech}}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C$

$\int {\mbox{sech}}\,x\,dx=\arctan \,(\sinh x)+C$

$\int \coth x\,dx=\ln |\sinh x|+C$

$\int {\mbox{sech}}^{2}x\,dx=\tanh x+C$

Inverse hyperbolic functions

more integrals: List of integrals of inverse hyperbolic functions

$\int \operatorname {arsinh} \,x\,dx=x\,\operatorname {arsinh} \,x-{\sqrt {x^{2}+1}}+C$

$\int \operatorname {arcosh} \,x\,dx=x\,\operatorname {arcosh} \,x-{\sqrt {x^{2}-1}}+C$

$\int \operatorname {artanh} \,x\,dx=x\,\operatorname {artanh} \,x+{\frac {1}{2}}\ln {(1-x^{2})}+C$

$\int \operatorname {arcsch} \,x\,dx=x\,\operatorname {arcsch} \,x+\ln {\left[x\left({\sqrt {1+{\frac {1}{x^{2}}}}}+1\right)\right]}+C$

$\int \operatorname {arsech} \,x\,dx=x\,\operatorname {arsech} \,x-\arctan {\left({\frac {x}{x-1}}{\sqrt {\frac {1-x}{1+x}}}\right)}+C$

$\int \operatorname {arcoth} \,x\,dx=x\,\operatorname {arcoth} \,x+{\frac {1}{2}}\ln {(x^{2}-1)}+C$

Composed functions

$\int \cos ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(a\sin ax+b\cos ax\right)+C$

$\int \sin ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(b\sin ax-a\cos ax\right)+C$

$\int \cos ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(a\sin ax\,\cosh bx+b\cos ax\,\sinh bx\right)+C$

$\int \sin ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(b\sin ax\,\sinh bx-a\cos ax\,\cosh bx\right)+C$

Absolute value functions

$\int \left|(ax+b)^{n}\right|\,dx={(ax+b)^{n+2} \over a(n+1)\left|ax+b\right|}+C\,\,[\,n{\text{ is odd, and }}n\neq -1\,]$

$\int \left|\sin {ax}\right|\,dx={-1 \over a}\left|\sin {ax}\right|\cot {ax}+C$

$\int \left|\cos {ax}\right|\,dx={1 \over a}\left|\cos {ax}\right|\tan {ax}+C$

$\int \left|\tan {ax}\right|\,dx={\tan(ax)[-\ln \left|\cos {ax}\right|] \over a\left|\tan {ax}\right|}+C$

$\int \left|\csc {ax}\right|\,dx={-\ln \left|\csc {ax}+\cot {ax}\right|\sin {ax} \over a\left|\sin {ax}\right|}+C$

$\int \left|\sec {ax}\right|\,dx={\ln \left|\sec {ax}+\tan {ax}\right|\cos {ax} \over a\left|\cos {ax}\right|}+C$

$\int \left|\cot {ax}\right|\,dx={\tan(ax)[\ln \left|\sin {ax}\right|] \over a\left|\tan {ax}\right|}+C$

Special functions

$\int \operatorname {Ci} (x)dx=x\,\operatorname {Ci} (x)-\sin x$

$\int \operatorname {Si} (x)dx=x\,\operatorname {Si} (x)+\cos x$

$\int \operatorname {Ei} (x)dx=x\,\operatorname {Ei} (x)-e^{x}$

$\int \operatorname {li} (x)dx=x\,\operatorname {li} (x)-\operatorname {Ei} (2\ln x)$

$\int {\frac {\operatorname {li} (x)}{x}}\,dx=\ln x\,\operatorname {li} (x)-x$

$\int \operatorname {erf} (x)\,dx={\frac {e^{-x^{2}}}{\sqrt {\pi }}}+x\,{\text{erf}}(x)$