Mathematics

$$\int { { cosec }^{ 3 }x } dx$$


SOLUTION
$$\int \csc ^3xdx$$

substitute $$ u=\tan \left ( \dfrac{x}{2} \right )$$

$$\therefore \int \csc ^3xdx=\int \dfrac{(1+u^2)^2}{4u^3}du$$

$$\int \dfrac{(1+u^2)^2}{4u^3}du = \dfrac{1}{4}\int\left ( \dfrac{1}{u^3}+\dfrac{2}{u}+u \right ) du$$

$$= \dfrac{1}{4}\left [ -\dfrac{1}{2u^2}+2\ln u+\dfrac{u^2}{2} \right ]$$

$$= \dfrac{1}{4}\left [ -\dfrac{1}{2\tan ^2\left ( \frac{x}{2} \right )}+2\ln \tan \left ( \dfrac{x}{2} \right )+\dfrac{\tan \left ( \frac{x}{2} \right )^2}{2} \right ]+C$$
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Subjective Medium Published on 17th 09, 2020
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