Mathematics

Evaluate the following integral:
$$\int { \cfrac { \cot { x }  }{ \sqrt { \sin { x }  }  }  } dx$$


SOLUTION
$$\displaystyle\int{\dfrac{\cot{x}}{\sqrt{\sin{x}}}dx}$$

$$=\displaystyle\int{\dfrac{\cos{x}}{\sin{x}\sqrt{\sin{x}}}dx}$$

Let $$t=\sin{x}\Rightarrow\,dt=\cos{x}dx$$

$$=\displaystyle\int{\dfrac{dt}{t\sqrt{t}}}$$

$$=\displaystyle\int{\dfrac{dt}{{t}^{1+\frac{1}{2}}}}$$

$$=\displaystyle\int{\dfrac{dt}{{t}^{\frac{3}{2}}}}$$

$$=\displaystyle\int{{t}^{\frac{-3}{2}}dt}$$

$$=\dfrac{{t}^{\frac{-3}{2}+1}}{\dfrac{-3}{2}+1}+c$$

$$=\dfrac{{t}^{\frac{-1}{2}}}{\dfrac{-1}{2}}+c$$

$$=-\dfrac{2}{\sqrt{t}}+c$$

$$=-\dfrac{2}{\sqrt{\sin{x}}}+c$$ where $$t=\sin{x}$$

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Subjective Medium Published on 17th 09, 2020
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