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
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