Using Partial Integration

Use partial integration to find:

1. \(\int_{0}^{4}\left[ \int_{1}^{2} x^{3}y^2\,{dx}\right] \!{\it dy}\)
2. \(\int_{1}^{2}\left[ \int_{0}^{4} x^{3}y^2\,{\it dy}\right] \! {dx}\)

Solution (a) We begin by using partial integration to find the integral within the bracket. Then we integrate the result with respect to \( y.\)

(b)