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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