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  4. ∫ log √π / 2 log √π e2 x sec 2((1/3) e2 x) d x is equal to

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Q. $\int_{\log \sqrt{\pi / 2}}^{\log \sqrt{\pi}} e^{2 x} \sec ^{2}\left(\frac{1}{3} e^{2 x}\right) d x$ is equal to

Integrals

Solution:

$
I=\int_{\log \sqrt{\pi / 2}}^{\log \sqrt{\pi}} e^{2 x} \sec ^{2}\left(\frac{1}{3} e^{2 x}\right) d x
$
Put $e^{2 x}=t \Rightarrow 2 e^{2 x} d x=d t$
When $x=\log \sqrt{\pi / 2}, t=e^{2 \log \sqrt{\pi / 2}}=e^{\log \pi / 2}=\frac{\pi}{2}$
When $x=\log \sqrt{\pi}, t=e^{2 \log \sqrt{\pi}}=\pi$
$
\begin{array}{l}
\therefore I=\int_{\frac{\pi}{2}}^{\pi} \frac{1}{2} \sec ^{2}\left(\frac{1}{3} t\right) d t=\frac{1}{2} \cdot \frac{1}{\frac{1}{3}}\left[\tan \frac{t}{3} \cdot\right]_{\pi / 2}^{\pi} \\
=\frac{3}{2}\left[\tan \frac{\pi}{3}-\tan \frac{\pi}{6}\right]=\frac{3}{2}\left[\sqrt{3}-\frac{1}{\sqrt{3}}\right]=\sqrt{3}
\end{array}
$