Mathematics

Evaluate the given integral.
$$\int { { e }^{ x } } \left( \cot { x } -co\sec ^{ 2 }{ x }  \right) dx$$


SOLUTION
$$I=\displaystyle\int{{e}^{x}\left(\cot{x}-{cosec}^{2}{x}\right)dx}$$

$$I=\displaystyle\int{{e}^{x}\cot{x}\,dx}-\displaystyle\int{{e}^{x}{cosec}^{2}{x}\,dx}$$

$$u={e}^{x}\Rightarrow\,du={e}^{x}\,dx$$

$$dv={cosec}^{2}{x}\,dx\Rightarrow\,v=-\cot{x}$$

$$\int u.v dx=u \int vdx-\int \left [\int vdx. \dfrac{du}{dx}.dx \right ] $$......by parts formula.

$$\Rightarrow\,I=\displaystyle\int{{e}^{x}\cot{x}\,dx}-\left[-{e}^{x}\cot{x}-\displaystyle\int{-{e}^{x}\cot{x}dx}\right]+c$$

$$\Rightarrow\,I=\displaystyle\int{{e}^{x}\cot{x}\,dx}+{e}^{x}\cot{x}-\displaystyle\int{{e}^{x}\cot{x}dx}+c$$

$$\therefore\,I={e}^{x}\cot{x}+c$$
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