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

\begin{align} \displaystyle \int{\cos{(2x+4)}}dx &= \dfrac{1}{2}\sin{(2x+4)} +c \\ \end{align}

### Example 2

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

\begin{align} \displaystyle \int{6\cos{\dfrac{4x}{3}}}dx &= 6 \times \dfrac{1}{\frac{4}{3}}\sin{\dfrac{4x}{3}} +c\\ &= \dfrac{9}{2}\sin{\dfrac{4x}{3}} +c \end{align}

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

\begin{align} \displaystyle \int{\sin{(4x-1)}}dx &= -\dfrac{1}{4}\cos{(4x-1)} +c \\ \end{align}

### Example 4

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

\begin{align} \displaystyle \int{-\sin{\dfrac{-4x}{\pi}}}dx &= -\dfrac{1}{\frac{-4}{\pi}}\sin{\dfrac{-4x}{\pi}} +c \\ &= \dfrac{\pi}{4}\sin{\dfrac{-4x}{\pi}} +c \end{align}

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