Question

Evaluate the integral of 4x^2(2+x^3)^9 dx.

I'm supposed to use the substitution rule but I have absolutely no idea where to start with this problem.

Answer 1

You just look at the expression and then try defining a function u(x) such that in terms of u the integral will simplify. There is no "wrong choice" because the integral in terms of u will always be the same as the original one. However, the goal is to evaluate the integral, so it's only useful when it simplifies.

In this case, the (1+x^3)^9 factor makes the integral complicated. If you define:

u(x) = 1+x^3

then

du = (du/dx) dx = 3x^2 dx

Therefore:

4x^2(2+x^3)^9 dx =

(2+x^3)^9 4/3 3 x^2 dx =

4/3 u^9 du

I think you can integrate this easily.