Mathematics

Solve $$\int {\frac{1}{\cos ^2x(1-\tan x)^2}dx}$$


SOLUTION
$$ I = \int \frac{1}{cos^{2}x(1-tanx)^{2}}dx$$
$$ I = \int \frac{sec^{2}x}{(1-tanx)^{2}}dx$$
$$ 1-tanx = t$$
$$ -sec^{2}xdx = dt$$
$$ I = -\int \frac{dt}{t^{2}}$$
$$ = -\frac{t^{-2+1}}{(-2+1)}+c$$
$$ I = +t^{-1}+c$$
$$ I = \frac{1}{t}+c$$
$$ I = \frac{1}{(1-tanx)}+c$$
$$ \therefore \int \frac{1}{cos^{2}x(1-tanx)^{2}}dx = \frac{1}{(1-tanx)}+c$$
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Subjective Medium Published on 17th 09, 2020
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