Editing Integration by parts (section)

====Polynomials and trigonometric functions====

In order to calculate

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

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

then:

<math display="block">\begin{align}
  \int x\cos(x)\,dx & = \int u\ dv \\
  & = u\cdot v - \int v \, du \\
  & = x\sin(x) - \int \sin(x)\,dx \\
  & = x\sin(x) + \cos(x) + C,
 \end{align}</math>

where ''C'' is a [[constant of integration]].

For higher powers of <math>x</math> in the form

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

repeatedly using integration by parts can evaluate integrals such as these; each application of the theorem lowers the power of <math>x</math> by one.