Question

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

• A. a multiple of $\pi /4$
• B. a multiple of $\pi /2$
• C. equal to $\pi /4$
• D. a multiple of $\pi$

Answer 1

Answer: (A), (C)

The correct options are
A a multiple of $\pi /4$
C equal to $\pi /4$

$I={\int }_0^{\pi /2}\frac{dx}{1+\tan x}$
$={\int }_0^{\pi /2}\frac{dx}{1+\tan (\pi /2-x)}$(Property 8)
$={\int }_0^{\pi /2}\frac{dx}{(1-\cot x}={\int }_0^{\pi /2}\frac{\tan x}{1+\tan x}dx$
$={\int }_0^{\pi /2}(1-\frac{1}{1+\tan x})dx=\frac{\pi }{2}-I$

$\Rightarrow I=\pi /4$