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 expression}\\ \text{in u} \end{matrix}\right)\cdot u'\,dx\,,

where \displaystyle u=u(x) 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 \bigl(x^2\bigr)'=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\,\textrm{.}

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

\displaystyle \begin{align} \int 2x\sin x^2\,dx &=\left\{\begin{align} u &= x^2\\[5pt] du &= 2x\,dx \end{align}\right\} = \int{\sin u\,du}\\[5pt] &= -\cos u+C = -\cos x^2 + C\,\textrm{.} \end{align}