Mathematics

# Evaluate the following definite integral:$\displaystyle\int_{0}^{\pi/4}\dfrac{\tan^{3}x}{1+\cos 2x}dx$

##### SOLUTION

Consider, $I=\displaystyle\int_{0}^{\pi/4}\dfrac{\tan^{3}x}{1+\cos 2x}dx$

$I=\displaystyle\int_{0}^{\pi/4}\dfrac{\tan^{3}x}{2\cos^{2}x}dx$

$=\dfrac{1}{2}\displaystyle\int_{0}^{\pi/4}\tan^{3}x\sec^{2}x\ dx$

$=\dfrac{1}{2}\displaystyle\int_{0}^{1}t^{3}dt$, where $t=\tan x$

$\Rightarrow I=\dfrac{1}{2}\left[\dfrac{t^{4}}{4}\right]_{0}^{1}$

$=\dfrac{1}{2}\left(\dfrac{1}{4}-0\right)=\dfrac{1}{8}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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