Editing Lists of integrals (section)

==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 [[antiderivative]]s.

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

===Integrals with a singularity===
When there is a [[Singularity (mathematics)|singularity]] in the function being integrated such that the antiderivative becomes undefined at some point (the singularity), then ''C'' does not need to be the same on both sides of the singularity. The forms below normally assume the [[Cauchy principal value]] around a singularity in the value of ''C'', but this is not necessary in general. For instance, in
<math display="block">\int {1 \over x}\,dx = \ln \left|x \right| + C</math>
there is a singularity at 0 and the [[antiderivative]] becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −''i''{{pi}} when using a path above the origin and ''i''{{pi}} for a path below the origin. A function on the real line could use a completely different value of ''C'' on either side of the origin as in:<ref>[[Serge Lang]] . ''A First Course in Calculus'', 5th edition, p. 290</ref>
<math display="block"> \int {1 \over x}\,dx = \ln|x| + \begin{cases} A & \text{if }x>0; \\ B & \text{if }x < 0. \end{cases} </math>

===Rational functions===
{{see also|List of integrals of rational functions}}

*<math>\int a\,dx = ax + C</math>

The following function has a non-integrable singularity at 0 for {{math|''n'' ≤ −1}}:
*<math>\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \qquad\text{(for } n\neq -1\text{)}</math> ([[Cavalieri's quadrature formula]])
*<math>\int (ax + b)^n \, dx= \frac{(ax + b)^{n+1}}{a(n + 1)} + C \qquad\text{(for } n\neq -1\text{)}</math>
*<math>\int {1 \over x}\,dx = \ln \left|x \right| + C</math>
**More generally,<ref>"[http://golem.ph.utexas.edu/category/2012/03/reader_survey_logx_c.html Reader Survey: log|''x''| + ''C'']", Tom Leinster, ''The ''n''-category Café'', March 19, 2012</ref><math display="block">\int {1 \over x}\,dx = \begin{cases}
\ln \left|x \right| + C^- & x < 0\\
\ln \left|x \right| + C^+ & x > 0
\end{cases}</math>
*<math>\int\frac{c}{ax + b} \, dx= \frac{c}{a}\ln\left|ax + b\right| + C</math>

===Exponential functions===
{{see also|List of integrals of exponential functions}}
*<math>\int e^{ax}\,dx = \frac{1}{a}e^{ax} + C</math>
*<math>\int f'(x)e^{f(x)}\,dx = e^{f(x)} + C</math>
*<math>\int a^x\,dx = \frac{a^x}{\ln a} + C</math>
*<math>\int{e^{x}\left( f\left( x \right) + f'\left( x \right) \right)\,dx} = e^{x}f\left( x \right) + C</math>
*<math>\int {e^{x}\left( f\left( x \right) - \left( - 1 \right)^{n}\frac{d^{n}f\left( x \right)}{dx^{n}} \right)\,dx} = e^{x}\sum_{k = 1}^{n}{\left( - 1 \right)^{k - 1}\frac{d^{k - 1}f\left( x \right)}{dx^{k - 1}}} + C</math>{{pb}}(if <math>n</math> is a positive integer)
*<math>\int {e^{- x}\left( f\left( x \right) - \frac{d^{n}f\left( x \right)}{dx^{n}} \right)\, dx} = - e^{- x}\sum_{k = 1}^{n}\frac{d^{k - 1}f\left( x \right)}{dx^{k - 1}} + C</math>{{pb}}(if <math>n</math> is a positive integer)

===Logarithms===
{{see also|List of integrals of logarithmic functions}}
*<math>\int \ln x\,dx = x \ln x - x + C = x (\ln x - 1) + C</math>
*<math>\int \log_a x\,dx = x\log_a x - \frac{x}{\ln a} + C = \frac{x}{\ln a} (\ln x - 1) + C</math>

===Trigonometric functions===
{{see also|List of integrals of trigonometric functions}}

*<math>\int \sin{x}\, dx = -\cos{x} + C</math>
*<math>\int \cos{x}\, dx = \sin{x} + C</math>
*<math>\int \tan{x} \, dx = \ln{\left| \sec{x} \right|} + C = -\ln{\left| \cos {x} \right|} + C</math>
*<math>\int \cot{x} \, dx =  -\ln{\left| \csc{x} \right|} + C = \ln{\left| \sin{x} \right|} + C</math>
*<math>\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C = \ln\left| \tan\left(\dfrac{x}{2} + \dfrac{\pi}{4}\right) \right| + C</math>
** (See [[Integral of the secant function]]. This result was a well-known conjecture in the 17th century.)
*<math>\int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + C = \ln{\left| \csc{x} - \cot{x}\right|} + C = \ln{\left| \tan {\frac{x}{2}} \right|} + C </math>
*<math>\int \sec^2 x \, dx = \tan x + C</math>
*<math>\int \csc^2 x \, dx = -\cot x + C</math>
*<math>\int \sec{x} \, \tan{x} \, dx = \sec{x} + C</math>
*<math>\int \csc{x} \, \cot{x} \, dx = -\csc{x} + C</math>
*<math>\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 </math>
*<math>\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 </math>
*<math>\int \tan^2 x \, dx = \tan x - x + C</math>
*<math>\int \cot^2 x \, dx = -\cot x - x + C</math>
*<math>\int \sec^3 x \, dx = \frac{1}{2}(\sec x \tan x + \ln|\sec x + \tan x|) + C</math>
** (See [[integral of secant cubed]].)
*<math>\int \csc^3 x \, dx = \frac{1}{2}(-\csc x \cot x + \ln|\csc x - \cot x|) + C = \frac{1}{2}\left(\ln\left|\tan\frac{x}{2}\right| - \csc x \cot x \right) + C</math>
*<math>\int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx</math>
*<math>\int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx</math>

===Inverse trigonometric functions===
{{see also|List of integrals of inverse trigonometric functions}}

*<math>\int \arcsin{x} \, dx = x \arcsin{x} + \sqrt{1 - x^2} + C , \text{ for } \vert x \vert \le 1 </math>
*<math>\int \arccos{x} \, dx = x \arccos{x} - \sqrt{1 - x^2} + C , \text{ for } \vert x \vert \le 1 </math>
*<math>\int \arctan{x} \, dx = x \arctan{x} - \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C , \text{ for all real } x </math>
*<math>\int \arccot{x} \, dx = x \arccot{x} + \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C , \text{ for all real } x </math>
*<math>\int \arcsec{x} \, dx = x \arcsec{x} - \ln \left\vert x \, \left( 1 + \sqrt{ 1 - x^{-2} } \, \right) \right\vert + C , \text{ for } \vert x \vert \ge 1 </math>
*<math>\int \arccsc{x} \, dx = x \arccsc{x} + \ln \left\vert x \, \left( 1 + \sqrt{ 1 - x^{-2} } \, \right) \right\vert + C , \text{ for } \vert x \vert \ge 1 </math>

===Hyperbolic functions===
{{see also|List of integrals of hyperbolic functions}}
*<math>\int \sinh x \, dx = \cosh x + C</math>
*<math>\int \cosh x \, dx = \sinh x + C</math>
*<math>\int \tanh x \, dx = \ln\,(\cosh x) + C</math>
*<math>\int \coth x \, dx = \ln| \sinh x | + C , \text{ for } x \neq 0 </math>
*<math>\int \operatorname{sech}\,x \, dx = \arctan\,(\sinh x) + C</math>
*<math>\int \operatorname{csch}\,x \, dx = \ln|\operatorname{coth} x - \operatorname{csch} x| + C = \ln\left| \tanh {x \over2}\right| + C , \text{ for } x \neq 0 </math>
*<math>\int \operatorname{sech}^2 x \, dx = \tanh x + C</math>
*<math>\int \operatorname{csch}^2 x \, dx = -\operatorname{coth}x + C</math>
*<math>\int \operatorname{sech}{x} \, \operatorname{tanh}{x} \, dx = -\operatorname{sech}{x} + C</math>
*<math>\int \operatorname{csch}{x} \, \operatorname{coth}{x} \, dx = -\operatorname{csch}{x} + C</math>

===Inverse hyperbolic functions===
{{see also|List of integrals of inverse hyperbolic functions}}

*<math>\int \operatorname{arcsinh} \, x \, dx = x \, \operatorname{arcsinh} \, x - \sqrt{ x^2 + 1 } + C , \text{ for all real } x </math>
*<math>\int \operatorname{arccosh} \, x \, dx = x \, \operatorname{arccosh} \, x - \sqrt{ x^2 - 1 } + C , \text{ for } x \ge 1 </math>
*<math>\int \operatorname{arctanh} \, x \, dx = x \, \operatorname{arctanh} \, x + \frac{\ln\left(\,1-x^2\right)}{2} + C , \text{ for } \vert x \vert < 1 </math>
*<math>\int \operatorname{arccoth} \, x \, dx = x \, \operatorname{arccoth} \, x + \frac{\ln\left(x^2-1\right)}{2} + C , \text{ for } \vert x \vert > 1 </math>
*<math>\int \operatorname{arcsech} \, x \, dx = x \, \operatorname{arcsech} \, x + \arcsin x + C , \text{ for } 0 < x \le 1 </math>
*<math>\int \operatorname{arccsch} \, x \, dx = x \, \operatorname{arccsch} \, x + \vert \operatorname{arcsinh} \, x \vert + C , \text{ for } x \neq 0 </math>

===Products of functions proportional to their second derivatives===
*<math>\int \cos ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( a\sin ax + b\cos ax \right) + C</math>
*<math>\int \sin ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( b\sin ax - a\cos ax \right) + C</math>
*<math>\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</math>
*<math>\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</math>

===Absolute-value functions===
Let {{math|''f''}} be a [[continuous function]], that has at most one [[zero of a function|zero]]. If {{math|''f''}} has a zero, let {{math|''g''}} be the unique antiderivative of {{math|''f''}} that is zero at the root of {{math|''f''}}; otherwise, let {{math|''g''}} be any antiderivative of {{math|''f''}}. Then
<math display="block">\int \left| f(x)\right|\,dx = \sgn(f(x))g(x)+C,</math>
where {{math|sgn(''x'')}} is the [[sign function]], which takes the values −1, 0, 1 when {{math|''x''}} is respectively negative, zero or positive.

This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on {{math|''g''}} is here for insuring the continuity of the integral.

This gives the following formulas (where {{math|''a'' ≠ 0}}), which are valid over any interval where {{math|''f''}} is continuous (over larger intervals, the constant {{mvar|C}} must be replaced by a [[piecewise constant]] function):
*<math>\int \left| (ax + b)^n \right|\,dx = \sgn(ax + b) {(ax + b)^{n+1} \over a(n+1)} + C</math>{{pb}}when {{math|''n''}} is odd, and <math>n \neq -1</math>.
*<math>\int \left| \tan{ax} \right|\,dx = -\frac{1}{a}\sgn(\tan{ax}) \ln(\left|\cos{ax}\right|) + C</math>{{pb}}when <math display="inline">ax \in \left( n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2} \right) </math> for some integer {{math|''n''}}.
*<math>\int \left| \csc{ax} \right|\,dx = -\frac{1}{a}\sgn(\csc{ax}) \ln(\left| \csc{ax} + \cot{ax} \right|) + C </math>{{pb}}when <math>ax \in \left( n\pi, n\pi + \pi \right) </math> for some integer {{math|''n''}}.
*<math>\int \left| \sec{ax} \right|\,dx = \frac{1}{a}\sgn(\sec{ax}) \ln(\left| \sec{ax} + \tan{ax} \right|) + C </math>{{pb}}when <math display="inline">ax \in \left( n\pi - \frac{\pi}{2}, n\pi + \frac{\pi}{2} \right) </math> for some integer {{math|''n''}}.
*<math>\int \left| \cot{ax} \right|\,dx = \frac{1}{a}\sgn(\cot{ax}) \ln(\left|\sin{ax}\right|) + C </math>{{pb}}when <math>ax \in \left( n\pi, n\pi + \pi \right) </math> for some integer {{math|''n''}}.

If the function {{math|''f''}} does not have any continuous antiderivative which takes the value zero at the zeros of {{math|''f''}} (this is the case for the sine and the cosine functions), then {{math|sgn(''f''(''x'')) ∫ ''f''(''x'') ''dx''}} is an antiderivative of {{math|''f''}} on every [[interval (mathematics)|interval]] on which {{math|''f''}} is not zero, but may be discontinuous at the points where {{math|1=''f''(''x'') = 0}}. For having a continuous antiderivative, one has thus to add a well chosen [[step function]]. If we also use the fact that the absolute values of sine and cosine are periodic with period {{pi}}, then we get:

*<math>\int \left| \sin{ax} \right|\,dx = {2 \over a} \left\lfloor \frac{ax}{\pi} \right\rfloor - {1 \over a} \cos{\left( ax - \left\lfloor \frac{ax}{\pi} \right\rfloor \pi \right)} + C</math> {{citation needed|date=April 2013}}
*<math>\int \left|\cos {ax}\right|\,dx = {2 \over a} \left\lfloor \frac{ax}{\pi} + \frac12 \right\rfloor + {1 \over a} \sin{\left( ax - \left\lfloor \frac{ax}{\pi} + \frac12 \right\rfloor \pi \right)} + C</math> {{citation needed|date=April 2013}}

===Special functions===
{{math|Ci}}, {{math|Si}}: [[Trigonometric integral]]s, {{math|Ei}}: [[Exponential integral]], {{math|li}}: [[Logarithmic integral function]], {{math|erf}}: [[Error function]]
* <math>\int \operatorname{Ci}(x) \, dx = x \operatorname{Ci}(x) - \sin x</math>
* <math>\int \operatorname{Si}(x) \, dx = x \operatorname{Si}(x) + \cos x</math>
* <math>\int \operatorname{Ei}(x) \, dx = x \operatorname{Ei}(x) - e^x</math>
* <math>\int \operatorname{li}(x) \, dx = x \operatorname{li}(x)-\operatorname{Ei}(2 \ln x) </math>
* <math>\int \frac{\operatorname{li}(x)}{x}\,dx = \ln x\, \operatorname{li}(x) -x </math>
* <math>\int \operatorname{erf}(x)\, dx = \frac{e^{-x^2}}{\sqrt{\pi }}+x \operatorname{erf}(x)</math>