Using Partial Integration

Use partial integration to find:

1. \(\int_{1}^{2} x^{3}y^2\,{ dx}\)
2. \(\int_{0}^{4}x^{3}y^2\, {dy}\)

Solution (a) For \(\int_{1}^{2} x^{3}y^2\,{\it dx}\), we treat \(y\) as a constant and integrate with respect to \(x\): \[ \int_{1}^{2} x^{3}y^2\,{\it dx}=y^2\int_{1}^{2}x^{3}{\it dx}=y^2\left[ \dfrac{x^{4}}{4}\right] _{1}^{2}= y^2\left( 4-\dfrac{1}{4}\right) =\dfrac{15}{4}y^2 \]

(b) For \(\int_{0}^{4} x^{3}y^2\, {\it dy},\) we treat \(x\) as a constant and integrate with respect to \(y\): \[ \int_{0}^{4} x^{3}y^2\, {\it dy}= x^{3}\!\int_{0}^{4}y^2\, {\it dy}= x^{3}\left[ {\dfrac{y^{3}}{3}} \right] _{0}^{4}=\dfrac{64x^{3}}{3} \]