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