Mathematics

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


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

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

$$\Rightarrow$$ $$I=\dfrac{1}{2}\displaystyle\int_{0}^{1}t^{3}dt$$                 where $$t=\tan x \ \  dt=sec^2x \ dx $$

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

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

$$\Rightarrow$$ $$I=\dfrac{1}{8}$$

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Subjective Medium Published on 17th 09, 2020
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