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Q. The integral $\int sec^{2/3} x cosec^{4/3}x\, dx$ is equal to (Hence $C$ is a constant of integration)

JEE MainJEE Main 2019Integrals

Solution:

$I = \int \frac{dx}{(sin x)^{4/3} . (cos x)^{2/3}}$
$I = \int \frac{dx}{\bigg(\frac{sin x}{cos x}\bigg)^{4/3} . cos^2 x}$
$\Rightarrow \, I = \int \frac{sec^2 x}{(tan x)^{4/3}} dx$
put tanx = t $\Rightarrow \, \, sec^2 x dx = dt$
$\therefore \, \, I = \int \frac{dt}{t^{4/3}} \Rightarrow \, I = \frac{-3}{t^{1/3}} +c$
$\Rightarrow \, \, \, I = \frac{-3}{(tan x)^{1/3}}+c$