Mathematics

# Calculate the following integral:$\displaystyle\, \int_{\cfrac{\pi}{8}}^{\cfrac{\pi}{4}}\, \cot^2\, 2x\, dx$

##### SOLUTION
$\displaystyle\int_{\frac {\pi}{8}}^{\frac {\pi}{4}} \cot^{2} (2x)dx = \int_{\frac {\pi}{8}}^{\frac {\pi}{4}} -1 + \csc^{2} (2x)dx = -\int_{\frac {\pi}{8}}^{\frac {\pi}{4}} dx + \int_{\frac {\pi}{8}}^{\frac {\pi}{4}} \csc^{2} (2x) dx$
Apply substitution $u= 2x\ \quad \frac {1}{2}du = dx$

$\displaystyle= -\left [\frac {\pi}{4} - \frac {\pi}{8}\right ] + \int_{\frac {\pi}{4}}^{\frac {\pi}{2}} csc^{2}(u) \left (\frac {1}{2} du\right )$

$\displaystyle= \frac {-\pi}{8} + \frac {1}{2}\int_{\frac {\pi}{4}}^{\frac {\pi}{2}} csc^{2} (u) du$

$\displaystyle= \frac {-\pi}{8} + \frac {1}{2} [-\cot u|_{\frac {\pi}{4}}^{\frac {\pi}{2}}]$

$= \frac {-\pi}{8} + \frac {1}{2} \left [-\cot \frac {\pi}{2} + \cot \frac {\pi}{4}\right ]$

$= \dfrac {1}{2} - \dfrac {\pi}{8}$

$= \left (\dfrac {4 - \pi}{8}\right )$.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 128

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