Chapter 7 - Principles Of Integral Evaluation - 7.2 Integration By Parts - Exercises Set 7.2 - Page 499: 61

$$\eqalign{ & \left( {\text{a}} \right) - \frac{1}{4}{\sin ^3}x\cos x - \frac{3}{8}\sin x\cos x + \frac{3}{8}x + C \cr & \left( {\text{b}} \right)\frac{8}{{15}} \cr} $$

$$\eqalign{ & \left( {\text{a}} \right)\int {{{\sin }^4}x} dx \cr & {\text{Use the reduction formula }} \cr & \int {{{\sin }^n}x} dx = - \frac{1}{n}{\sin ^{n - 1}}x\cos x + \frac{{n - 1}}{n}\int {{{\sin }^{n - 2}}xdx} \cr & n = 4 \cr & \int {{{\sin }^4}x} dx = - \frac{1}{4}{\sin ^3}x\cos x + \frac{3}{4}\int {{{\sin }^2}xdx} \cr & n = 2 \cr & \int {{{\sin }^4}x} dx = - \frac{1}{4}{\sin ^3}x\cos x + \frac{3}{4}\left( { - \frac{1}{2}\sin x\cos x + \frac{1}{2}\int {dx} } \right) \cr & \int {{{\sin }^4}x} dx = - \frac{1}{4}{\sin ^3}x\cos x - \frac{3}{8}\sin x\cos x + \frac{3}{8}\int {dx} \cr & \int {{{\sin }^4}x} dx = - \frac{1}{4}{\sin ^3}x\cos x - \frac{3}{8}\sin x\cos x + \frac{3}{8}x + C \cr & \cr & \left( {\text{b}} \right)\int_0^{\pi /2} {{{\sin }^5}x} dx \cr & {\text{Use the reduction formula }} \cr & \int {{{\sin }^n}x} dx = - \frac{1}{n}{\sin ^{n - 1}}x\cos x + \frac{{n - 1}}{n}\int {{{\sin }^{n - 2}}xdx} \cr & n = 5 \cr & \int {{{\sin }^5}x} dx = - \frac{1}{4}{\sin ^4}x\cos x + \frac{4}{5}\int {{{\sin }^3}xdx} \cr & n = 3 \cr & \int {{{\sin }^5}x} dx = - \frac{1}{4}{\sin ^4}x\cos x + \frac{4}{5}\left( { - \frac{1}{3}{{\sin }^2}x\cos x + \frac{2}{3}\int {\sin xdx} } \right) \cr & \int {{{\sin }^5}x} dx = - \frac{1}{4}{\sin ^4}x\cos x - \frac{4}{{15}}{\sin ^2}x\cos x + \frac{8}{{15}}\int {\sin xdx} \cr & \int {{{\sin }^5}x} dx = - \frac{1}{4}{\sin ^4}x\cos x - \frac{4}{{15}}{\sin ^2}x\cos x - \frac{8}{{15}}\cos x + C \cr & \int_0^{\pi /2} {{{\sin }^5}x} dx = \left[ { - \frac{1}{4}{{\sin }^4}x\cos x - \frac{4}{{15}}{{\sin }^2}x\cos x - \frac{8}{{15}}\cos x} \right]_0^{\pi /2} \cr & {\text{Evaluating}} \cr & = \left[ { - \frac{1}{4}{{\sin }^4}\left( {\frac{\pi }{2}} \right)\cos \left( {\frac{\pi }{2}} \right) - \frac{4}{{15}}{{\sin }^2}\left( {\frac{\pi }{2}} \right)\cos \left( {\frac{\pi }{2}} \right) - \frac{8}{{15}}\cos \left( {\frac{\pi }{2}} \right)} \right] \cr & - \left[ { - \frac{1}{4}{{\sin }^4}\left( 0 \right)\cos \left( 0 \right) - \frac{4}{{15}}{{\sin }^2}\left( 0 \right)\cos \left( 0 \right) - \frac{8}{{15}}\cos \left( 0 \right)} \right] \cr & {\text{Simplifying}} \cr & = \left[ 0 \right] - \left[ {0 - 0 - \frac{8}{{15}}\cos \left( 0 \right)} \right] \cr & = \frac{8}{{15}} \cr} $$