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EXAMPLE 3Using Partial Integration

Use partial integration to find:

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

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

(b)