Evaluate $\int \sin^5 x \; \cos^4 x \; dx$
Let $I = \int \sin^5 x \; \cos^4 x \; dx $ $\;\;\; \cdots$ (1)
Compare equation (1) with $\int \sin^m x \; \cos^n x \; dx$
Here $m = 5$ (odd) and $n = 4$ (even)
$\therefore$ $\;$ Let $\cos x = p$ $\;\;\; \cdots$ (2a)
Differentiating equation (2a) gives
$- \sin x \; dx = dp$ $\;\;\; \cdots$ (2b)
Now, equation (1) can be rewritten as
$\begin{aligned}
I & = \int \sin x \cdot \sin^4 x \cdot \cos^4 x \; dx \\\\
& = \int \sin x \cdot \left(\sin^2 x\right)^2 \cdot \cos^4 x \; dx \\\\
& = \int \sin x \cdot \left(1 - \cos^2 x\right)^2 \cdot \cos^4 x \; dx \\\\
& = - \int \left(1 - p^2\right)^2 \cdot p^4 \; dp \;\;\; \left[\text{In view of equations (2a) and (2b)}\right] \\\\
& = - \int \left(1 - 2 p^2 + p^4\right) \cdot p^4 \; dp \\\\
& = \int \left(- p^4 + 2 p^6 - p^8 \right) \; dp \\\\
& = - \int p^4 \; dp + 2 \int p^6 \; dp - \int p^8 \; dp \\\\
& = - \dfrac{p^5}{5} + \dfrac{2 \; p^7}{7} - \dfrac{p^9}{p^9} + c \\\\
& = \dfrac{- \cos^5 x}{5} + \dfrac{2 \cos^7 x}{7} - \dfrac{\cos^9 x}{9} + c \;\;\; \left[\text{In view of equation (2a)}\right]
\end{aligned}$