Editing Lists of integrals (section)

===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}}