# INTEGRATION OF TRIGONOMETRIC

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When differentiating trigonometric function, we note the following:

$$\frac{d}{dx}(\sin x)=\cos x$$

$$\frac{d}{dx}(\cos x)=-\sin x$$

$$\frac{d}{dx}(\tan x)=\sec^2 x$$

$$\frac{d}{dx}(\sec x)=\sec x \tan x$$

$$\frac{d}{dx}(\csc x)=-\csc x \cot x$$

$$\frac{d}{dx}(\cot x)=-\csc^2 x$$

Since integration is the inverse of differentiation, it follows that

$$\int \cos x dx=\sin x+C$$

$$\int \sin x dx =-\cos x+C$$

$$\int \sec^2 x dx= \tan x+C$$

$$\int \csc x \cot x= -\csc x+C$$

$$\int \csc^2 =-\cot x+C$$

Example 1

Integrate $7\sec^2x$

Solution:

$\int 7\sec^2x=7\int\sec^2x$

$7(\tan x)+ C$

Example 2

Evaluate $\int \left(\cos x +7\sin x-\sec^2x\right) dx$

Solution:

$\sin x+ 7 (-\cos x)-\tan x + C$

$\sin x-7\cos x-\tan x + C$

Example 3

Evaluate $\int \left( 1+ \cos x +\csc^2 x\right) dx$

Solution:

Here, we use the power rule for 1 and we integrate the other using the normal procedure

$\frac{1x^{0+1}}{1}+\sin x +(-\cot x) +C$

$x+\sin x-\cot x +C$

Example 4

Integrate $1+2\cos x+9\csc x \cot x$

Solution:

$\int \left( 1+2\cos x+9\csc x \cot x\right) dx$

$1x+2\sin x +9 (-\csc x)+C$

$x+2\sin x -9\csc x+ C$

Example 5

Evaluate $\int \cos 3x dx$

Solution:

$\frac{1}{3}\sin 3x+C$

$\frac{\sin 3x}{3}+C$

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