Trigonometric Integration by Substitution

Substitution of $\sin{x}$

$$ \begin{align} \displaystyle \text{ Let} u &= \sin{x} \\ \dfrac{du}{dx} &= \cos{x} \\ du &= \cos{x}du \\ \int{\sin^n{x}\cos{x}}dx &= \int{u^n}du \\ &= \dfrac{u^{n+1}}{n+1} + c \\ &= \dfrac{\sin^{n+1}{x}}{n+1} + c \\ \end{align} $$

Substitution of $\cos{x}$

$$ \begin{align} \displaystyle \text{ Let} u &= \cos{x} \\ \dfrac{du}{dx} &= -\sin{x} \\ -du &= \sin{x}du \\ \int{\cos^n{x}\sin{x}}dx &= -\int{u^n}du \\ &= -\dfrac{u^{n+1}}{n+1} + c \\ &= -\dfrac{\cos^{n+1}{x}}{n+1} + c \\ \end{align} $$

Example 1

Find $\displaystyle \int{\sin^3{x}}dx$.

Example 2

Find $\displaystyle \int{\cos^5{x}\sin^4{x}}dx$.