Question

How do you integrate `int x^3 sin x^2 dx` using integration by parts?

Answer 1

`1/2(sin(x^2)-x^2cos(x^2))+C`

Explanation:

`d/dx(x^2cos(x^2))=2xcos(x^2)-2x^3sin(x^2)`

but `2xcos(x^2)= d/dx(sin(x^2))` so

`d/dx(x^2cos(x^2)) = d/dx(sin(x^2))-2x^3sin(x^2)`

then

`int x^3sin(x^2)dx = 1/2(sin(x^2)-x^2cos(x^2))+C`