Integrate the function: 1 ^ 2 x(1- x)^ 2 - Vidyadip

Question

Integrate the function: $\frac{1}{\cos ^{2} x(1-\tan x)^{2}}$

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Answer

We have $\frac{1}{\cos ^{2} x(1-\tan x)^{2}}=\frac{\sec ^{2} x}{(1-\tan x)^{2}}$
Let $(1 – \tan x) = t$
$\Rightarrow -\sec^2 x\ dx = dt$
$\therefore ~ \int \frac{\sec ^{2} x}{(1-\tan x)^{2}} d x=\int \frac{-d t}{t^2}=-\int t^{-2} d t$
$= \frac{1}{t}+C$
$= \frac{1}{(1-\tan x)}+C$

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