Editing Integration by parts (section)

====Exponentials and trigonometric functions====
{{hatnote|See also: [[Integration using Euler's formula]]}}
An example commonly used to examine the workings of integration by parts is

<math display="block">I=\int e^x\cos(x)\,dx.</math>

Here, integration by parts is performed twice. First let

<math display="block">\begin{alignat}{3}
   u &= \cos(x)\ &\Rightarrow\ &&du &= -\sin(x)\,dx \\
  dv &= e^x\,dx\ &\Rightarrow\ &&v  &= \int e^x\,dx = e^x
\end{alignat}</math>

then:

<math display="block">\int e^x\cos(x)\,dx = e^x\cos(x) + \int e^x\sin(x)\,dx.</math>

Now, to evaluate the remaining integral, we use integration by parts again, with:

<math display="block">\begin{alignat}{3}
   u &= \sin(x)\ &\Rightarrow\ &&du &= \cos(x)\,dx \\
  dv &= e^x\,dx\,&\Rightarrow\ && v &= \int e^x\,dx = e^x.
\end{alignat}</math>

Then:

<math display="block">\int e^x\sin(x)\,dx = e^x\sin(x) - \int e^x\cos(x)\,dx.</math>

Putting these together,

<math display="block">\int e^x\cos(x)\,dx = e^x\cos(x) + e^x\sin(x) - \int e^x\cos(x)\,dx.</math>

The same integral shows up on both sides of this equation. The integral can simply be added to both sides to get

<math display="block">2\int e^x\cos(x)\,dx = e^x\bigl[\sin(x)+\cos(x)\bigr] + C,</math>

which rearranges to

<math display="block">\int e^x\cos(x)\,dx = \frac{1}{2}e^x\bigl[\sin(x)+\cos(x)\bigr] + C'</math>

where again <math>C</math> (and <math>C' = \frac{C}{2}</math>) is a [[constant of integration]].

A similar method is used to find the [[integral of secant cubed]].