Integration of Trigonometric Functions

Integration of $\cos{x}$

$$ \begin{align} \displaystyle \dfrac{d}{dx}\sin{x} &= \cos{x} \\ \sin{x} &= \int{\cos{x}}dx \\ \therefore \int{\cos{x}}dx &= \sin{x} +c \\ \end{align} $$

Integration of $\cos{(ax+b)}$

$$ \begin{align} \displaystyle \dfrac{d}{dx}\sin{(ax+b)} &= \cos{(ax+b)} \times \dfrac{d}{dx}(ax+b) \\ &= \cos{(ax+b)} \times a \\ &= a\cos{(ax+b)} \\ \sin{(ax+b)} &= \int{a\cos{(ax+b)}}dx \\ &= a\int{\cos{(ax+b)}}dx \\ \dfrac{1}{a}\sin{(ax+b)} &= \int{\cos{(ax+b)}}dx \\ \therefore \int{\cos{(ax+b)}}dx &= \dfrac{1}{a}\sin{(ax+b)} +c \\ \end{align} $$

Example 1

Find $\displaystyle \int{\cos{(2x+4)}}dx$.

Example 2

Find $\displaystyle \int{6\cos{\dfrac{4x}{3}}}dx$.

Integration of $\sin{x}$

$$ \begin{align} \displaystyle \dfrac{d}{dx}\cos{x} &= -\sin{x} \\ \cos{x} &= -\int{\sin{x}}dx \\ -\cos{x} &= \int{\sin{x}}dx \\ \therefore \int{\sin{x}}dx &= -\cos{x} +c \\ \end{align} $$

Integration of $\sin{(ax+b)}$

$$ \begin{align} \displaystyle \dfrac{d}{dx}\cos{(ax+b)} &= -\sin{(ax+b)} \times \dfrac{d}{dx}(ax+b) \\ &= -\sin{(ax+b)} \times a \\ &= -a\sin{(ax+b)} \\ \cos{(ax+b)} &= -\int{a\sin{(ax+b)}}dx \\ &= -a\int{\sin{(ax+b)}}dx \\ -\dfrac{1}{a}\cos{(ax+b)} &= \int{\sin{(ax+b)}}dx \\ \therefore \int{\sin{(ax+b)}}dx &= -\dfrac{1}{a}\cos{(ax+b)} +c \\ \end{align} $$

Example 3

Find $\displaystyle \int{\sin{(4x-1)}}dx$.

Example 4

Find $\displaystyle \int{-\sin{\dfrac{-4x}{\pi}}}dx$.