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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

an expressionin uudx

where u=u(x) is the actual substitution. In the integral

2xsinx2dx

we see that the expression x2 is the argument for the sine function, as the same time as its derivative x2=2x  stands as a factor in front of sine. Therefore, if we set u=x2, the integral, the integral will be of the form

usinudx.

Thus, we can use u=x2 for the substitution,

2xsinx2dx=udu=x2=2xdx=sinudu=−cosu+C=−cosx2+C.