Question

Evaluate: $\int cose{c}^{2}x\sqrt{\cot x.}dx$

• A. $-\frac{1}{3}{\left(\cot x\right)}^{3/2}.$
• B. $\frac{2}{3}{\left(\cot x\right)}^{3/2}.$
• C. $-\frac{2}{3}{\left(\cot x\right)}^{3/2}.$
• D. $-\frac{2}{3}{\left(\tan x\right)}^{3/2}.$

Answer 1

The correct option is C $-\frac{2}{3}{\left(\cot x\right)}^{3/2}.$

Let $I=\int co{\sec }^{2}x\sqrt{\cot x}$
Put $\cot x=t\Rightarrow -\cos x{\sec }^{2}xdx=dt$
Therefore
$I=-\int \sqrt{t}dt=-\frac{2}{3}{t}^{3/2}=-\frac{2}{3}{\left(\cot x\right)}^{3/2}$
Hence, option 'C' is correct.