Evaluate the following integrals: ^3x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\cos^3\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$

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Answer

$\int\frac{\cos^3\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$=\int\frac{\cos^2\text{x}\cos\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$=\int\frac{(1-\sin^2\text{x})\cos\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$\text{Let }\sin\text{x}=\text{t}$
$\Rightarrow\cos\text{x}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\cos\text{x dx}=\text{dt}$
$\text{Now,}\int\frac{(1-\sin^2\text{x})\cos\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$
$=\int\frac{(1-\text{t}^2)}{\sqrt{\text{t}}}\text{dt}$
$=\int\Big(\frac{1}{\sqrt{\text{t}}}-\text{t}^\frac{3}{2}\Big)\text{dt}$
$=\int\Big(\text{t}^{-\frac{1}{2}}-\text{t}^\frac{3}{2}\Big)\text{dt}$
$=\Bigg[\frac{\text{t}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}-\frac{\text{t}^{\frac{3}{2}+1}}{\frac{3}{2}+1}\Bigg]+\text{C}$
$=2\sqrt{\text{t}}-\frac{2}{5}\text{t}^\frac{5}{2}+\text{C}$
$=2\sqrt{\sin\text{x}}-\frac{2}{5}\sin^\frac{5}{2}\text{x}+\text{C}$

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