Question

Evaluate the integral
${\int }_0^{\pi /2}\frac{1}{1+\cot x}dx$

• A. $\pi$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is C $\frac{\pi }{4}$

$I={\int }_0^{\frac{\pi }{2}}\frac{1}{1+\cot x}dx$

${\int }_o^{a}f\left(x\right)dx={\int }_o^{a}f(a-x)dx$

$I={\int }_0^{\frac{\pi }{2}}\frac{1}{1+\tan x}dx={\int }_0^{\frac{\pi }{2}}\frac{\cot x}{1+\cot x}dx$

$2I={\int }_0^{\frac{\pi }{2}}1\cdot dx$

$=\frac{\pi }{2}$

$2I=\frac{\pi }{2}$

$I=\frac{\pi }{4}$