Question

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

• A. multiple of $\frac{\pi }{4}$
• B. multiple $\frac{\pi }{2}$
• C. multiple of $\pi$
• D. none of these

Answer 1

The correct option is A multiple of $\frac{\pi }{4}$

${\int }_0^{\pi /2}\frac{d_x}{1+\tan x}$ = ${\int }_0^{\pi /2}\frac{\cos x}{\sin x+\cos x}dx$ -(1)
$I={\int }_0^{\pi /2}\frac{\sin x}{\cos x+\sin x}dx$ -(2)
Adding 1 & 2, we get
$2I=\pi /2$
$I=\pi /4$
So, multiple of $\pi /4$