Evaluate the following integrals: dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\cot\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$

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Answer

Let I $=\int\frac{\cot\text{x}}{\sqrt{\sin\text{x}}}\text{dx}\ .....(1)$
Let $\sin\text{x}=\text{t}$ then,
$\text{d}(\sin\text{x})=\text{dt}$
$\Rightarrow\cos\text{x}\text{ dx}=\text{dt}$
$\text{Now,}\text{I}=\int\frac{\cot\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$=\int\frac{\cot\text{x}}{\sin\text{x}\sqrt{\sin\text{x}}}\text{dx}$
$\int\frac{\cos\text{x}}{(\sin\text{x})^{\frac{3}{2}}}\text{dx}$
$\Rightarrow\ =\int\frac{\cos\text{x}}{(\sin\text{x})^\frac{3}{2}}\text{dx}\ ...(2)$
Putting $\sin\text{x}=\text{t}$ and $\cos\text{x}\text{ dx}=\text{dt}$ in equation (2), we get
$\text{I}=\int\frac{\text{dt}}{\text{t}^\frac{3}{2}}$
$=\int\text{t}^{-\frac{3}{2}}\text{dt}$
$=-2\text{t}^{-\frac{1}{2}}+\text{C}$
$=\frac{-2}{\sqrt{\text{t}}}+\text{C}$
$=\frac{-2}{\sqrt{\sin\text{x}}}+\text{C}$
$\therefore\text{I}=\frac{-2}{\sqrt{\sin\text{x}}}+\text{C}$

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