Aus Online Mathematik Brückenkurs 2

If we to succeed in simplifying the integral with a substitution, we must find an expression \displaystyle u=u\left( x \right) so that the integral can be written as

\displaystyle \int{\left( \begin{matrix} \text{something} \\ \text{in}\quad u \\ \end{matrix} \right)}\centerdot {u}'\,dx.

As our integral is written,

\displaystyle \int{\sin x\cos x\,dx}

we see that the second factor \displaystyle \cos x is a derivative of the first factor, \displaystyle \sin x. If \displaystyle u=\text{sin }x, the integral can thus be written as

\displaystyle \int{u\centerdot {u}'\,dx}

and this makes \displaystyle u=\text{sin }x an appropriate substitution,

\displaystyle \begin{align} & \int{\sin x\cos x\,dx}=\left\{ \begin{matrix} u=\text{sin }x \\ du=\left( \sin x \right)^{\prime }\,dx=\cos x\,dx \\ \end{matrix} \right\} \\ & =\int{u\,du=\frac{1}{2}u^{2}} \\ & =\frac{1}{2}\sin ^{2}x+C \\ \end{align}