Aus Online Mathematik Brückenkurs 2

The secret behind a successful substitution is to be able to recognize the integral as an expression of the type

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

where \displaystyle u=u\left( x \right) is the actual substitution. In the integral

\displaystyle \int{2x\sin x^{2}\,dx}

we see that the expression \displaystyle x^{2} is the argument for the sine function, as the same time as its derivative \displaystyle \left( x^{2} \right)^{\prime }=2x stands as a factor in front of sine. Therefore, if we set \displaystyle u=x^{2}, the integral, the integral will be of the form

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

Thus, we can use \displaystyle u=x^{2} for the substitution:

\displaystyle \begin{align} & \int{2x\sin x^{2}\,dx}=\left\{ \begin{matrix} u=x^{2} \\ du=2x\,dx \\ \end{matrix} \right\}=\int{\sin u\,du} \\ & =-\cos u+C=-\cos x^{2}+C \\ \end{align}