$\int \sin^{5} x \cos^{4} x d x =$

Moderate

A

$- \frac{\sin^{4} x \cos^{5} x}{9} + \frac{4 \sin^{2} x \cos^{5} x}{63} + \frac{\cos^{5} x}{315} + C$
B

$\frac{\sin^{4} x \cos^{5} x}{9} + \frac{4 \sin^{2} x \cos^{5} x}{63} + \frac{\cos^{5} x}{315} + C$
C

$\frac{\sin^{4} x \cos^{5} x}{9} - \frac{4 \sin^{2} x \cos^{5} x}{63} + \frac{\cos^{5} x}{315} + C$
D

$- \frac{\sin^{4} x \cos^{5} x}{9} - \frac{4 \sin^{2} x \cos^{5} x}{63} - \frac{8 \cos^{5} x}{315} + C$

Solution

$I_{m, n} \int \sin^{m} x \cos^{n} x d x = \frac{\sin^{m + 1} x \cos^{n - 1}}{m + n} + \frac{n - 1}{m + n} I_{m, n - 2}$
$m = 5, n = 4$
$\begin{array}{l} = \frac{- \sin^{m - 1} x \cos^{n + 1} x}{m + n} + \frac{m - 1}{m + n} I_{m - 2, n} (m, n \in N, m \geq 2, n \geq 2) \\ I_{5} = \int \sin^{5} x \cos^{4} x d x = \frac{- \sin^{4} x \cos^{5} x}{9} + \frac{4}{9} I_{3, 4} \\ = \frac{- \sin^{4} x \cos^{5} x}{9} + \frac{4}{9} [\frac{- \sin^{2} x \cos^{5} x}{7} + \frac{2}{7} I_{1, 4}] \\ = \frac{- \sin^{4} x \cos^{5} x}{9} - \frac{4}{63} \sin^{2} x \cos^{5} x + \frac{8}{63} \int \sin x \cos^{4} x d x \\ = \frac{- \sin^{4} x \cos^{5} x}{9} - \frac{4}{63} \sin^{2} x \cos^{5} x - \frac{8 \cos^{5} x}{315} + C \end{array}$

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