# Find the indefinite integral of the following functions (a) (tan(x))^{?3/2} sec^2(x) (b)...

## Question:

Find the indefinite integral of the following functions

(a) {eq}(\tan(x))^{-\frac{3}{2}}\sec^2(x) {/eq}

(b) {eq}\tan^3(x)(\sec(x))^{-\frac12} {/eq}

(c) {eq}\tan^5(x)\sec^9(x) {/eq}

## Integrals of Trigonometric Functions:

To evaluate the integrals of trigonometric functions, it is required to find a relationship between the trigonometric functions. Any identities or relationship between the trigonometric functions may assist to simplify the expression under integral.

In this problem, the identity {eq}\sec^2\theta = \tan^2\theta +1\\ {/eq} or {eq}\tan^2\theta =\sec^2\theta -1\\ {/eq}

(a) {eq}(\tan(x))^{-\frac{3}{2}}\sec^2(x) {/eq}

$$\int (\tan(x))^{-\frac{3}{2}}\sec^2(x) dx\\$$

By substitution method

u=\tan(x)\\ du=sec^2(x...

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