Question

Solve $\underset{0}{\overset{\pi /4}{\int }}{\left(\frac{1}{\sin x}\right)}^{2}dx$

• A. $-\pi$
• B. $-1$
• C. $\pi$
• D. $1$

Answer 1

The correct option is B $-1$

Consider the given integral.

$I={{\int }_0^{\frac{\pi }{4}}\left(\frac{1}{\sin x}\right)}^{2}dx$

$I={\int }_0^{\frac{\pi }{4}}{\left(\csc x\right)}^{2}dx$

$I={\int }_0^{\frac{\pi }{4}}\left({\csc }^{2}x\right)dx$

$I={[-\cot x]}_0^{\frac{\pi }{4}}$

$I=-(\cot \frac{\pi }{4}-\cot 0)$

$I=-(1-0)$

$I=-1$

Hence, this is the answer.