User:Barrucadu/ListOfIntegrals

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
 * $$\int \ln x\,dx = x \ln x - x + C$$
 * $$\int \log_a x\,dx = x\log_a x - \frac{x}{\ln a} + C$$

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
 * $$\int \arcsin{x} \, dx = x \, \arcsin{x} + \sqrt{1 - x^2} + C$$
 * $$\int \arccos{x} \, dx = x \, \arccos{x} - \sqrt{1 - x^2} + C$$
 * $$\int \arctan{x} \, dx = x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C$$
 * $$\int \arccot{x} \, dx = x \, \arccot{x} + \frac{1}{2} \ln{\left| 1 + x^2\right|} + C$$
 * $$\int \arcsec{x} \, dx = x \, \arcsec{x} - \operatorname{arcosh} \, x + C$$
 * $$\int \arccsc{x} \, dx = x \, \arccsc{x} + \operatorname{arcosh} \, x + C$$

Hyperbolic functions

 * more integrals: List of integrals of hyperbolic functions
 * $$\int \sinh x \, dx = \cosh x + C$$
 * $$\int \cosh x \, dx = \sinh x + C$$
 * $$\int \tanh x \, dx = \ln| \cosh x | + C$$
 * $$\int \mbox{cosech}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C$$
 * $$\int \mbox{sech}\,x \, dx = \arctan\,(\sinh x) + C$$
 * $$\int \coth x \, dx = \ln| \sinh x | + C$$
 * $$\int \mbox{sech}^2 x\, dx = \tanh x + C$$

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

 * $$\int \operatorname{Ci}(x) dx = x\,\operatorname{Ci}(x) - \sin x$$
 * $$\int \operatorname{Si}(x) dx = x\,\operatorname{Si}(x) + \cos x$$
 * $$\int \operatorname{Ei}(x) dx = x\,\operatorname{Ei}(x) - e^x$$
 * $$\int \operatorname{li}(x)dx = x\, \operatorname{li}(x)-\operatorname{Ei}(2 \ln x) $$
 * $$\int \frac{\operatorname{li}(x)}{x}\,dx = \ln x\, \operatorname{li}(x) -x $$
 * $$\int \operatorname{erf}(x)\, dx = \frac{e^{-x^2}}{\sqrt{\pi }}+x\, \text{erf}(x)$$