Evaluate the following integrals: _ 0 ^ 4 ^ 3 x 1+ 2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{4}}\frac{\tan^{3}\text{x}}{1+\cos2\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int_{0}^\limits{\frac{\pi}{4}}\frac{\tan^{3}\text{x}}{1+\cos2\text{x}}\text{ dx}$ Then,
$\text{I}=\int_{0}^\limits{\frac{\pi}{4}}\frac{\tan^{3}\text{x}}{1+\cos2\text{x}}\text{ dx}$
$\Rightarrow\text{I}=\frac{1}{2}\int_{0}^\limits{\frac{\pi}{4}}\tan^{3}\text{x}\sec^2\text{x}\text{ dx}$
Let $\tan\text{x}=\text{t}$ Then, $\sec^2\text{x dx}=\text{dt}$
When $\text{x}=0,\text{t}=0$ and $\text{x}=\frac{\pi}{4},\text{t}=1$
$\therefore\ \text{I}=\frac{1}{2}\int_{0}^\limits{1}\text{t}^3\text{ d}t$
$\Rightarrow\text{I}=\frac{1}{2}\Big[\frac{\text{t}^4}{4}\Big]^1_0$
$\Rightarrow\text{I}=\frac{1}{2}\Big(\frac{1}{4}-0\Big)$
$\Rightarrow\text{I}=\frac{1}{8}$

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