Integrate the function xsin(4x^2) with respect to x, using the integration by substitution method.

Heres our integral: int(xsin(4x²)dx)The first decision to make is what to use as our substitution. We're going to go for the term tied up in the sine function: 4x²Set u = 4x²du/dx = 8x, therefore dx = du/8xSubstituting this into our problem gives:int(xsin(u)du/8x) = (1/8)int(sin(u)du)This can now be solved using the integration identity for the sine function to give our answer:-cos(u)(1/8) + const = -cos(4x²)(1/8) + constOnce you have the method down, the key tricks to master are recognising when to use the substitution method (if the question doesn't tell you which method to use), and what term to use as your substitution. You'll get the hang of these firstly by solving lots of problems using this method and developing an intuition for it. Secondly by thinking carefully about what the substitution method allows you to do versus other methods, and the significance of the substituted term.