# Evaluate the integrals. A) Integral of tan^2 x sec^4 x dx. B) Integral of tan^2 x sec x dx.

## Question:

Evaluate the integrals.

A) {eq}\int \tan^2 x \sec^4 x \, \mathrm{d}x {/eq}

B) {eq}\int \tan^2 x \sec x \, \mathrm{d}x {/eq}

## Intergation:

The reverse process of differentiation is integration. It can be related by using fundamental theorem of calculus. By using substitution method complex integral can be reduced to standard from.

## Answer and Explanation:

{eq}\int \tan^{2}x\sec^{4}xdx\\ =\int \tan^{2}x(1+\tan^{2}x)\sec^{2}xdx\\ =\int t^{2}(1+t^{2})dt\\ =\frac{\tan^{3}x}{3}+\frac{\tan^{5}x}{5}+c\\ {/eq}

{eq}b)\int \left (\sec^{2}x-1 \right )\sec xdx\\ =\frac{\sec x \tan x}{2} -\frac{\ln |\sec x+\tan x|}{2} {/eq}

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