Question

The value of the integral $\underset{0}{\overset{\pi /2}{\int }}\frac{1}{1+(\tan x{)}^{101}}dx$ is equal to

• A. $1$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{8}$
• D. $\frac{\pi }{4}$

Answer 1

Answer: (D)

Let $I=\underset{0}{\overset{\pi /2}{\int }}\frac{1}{1+(\tan x{)}^{101}}dx$
$=\underset{0}{\overset{\pi /2}{\int }}\frac{dx}{1+{\left\{\tan \left(\frac{\pi }{2}-x\right)\right\}}^{101}}$
$=\underset{0}{\overset{\pi /2}{\int }}\frac{dx}{1+(\cot x{)}^{101}}$
$=\underset{0}{\overset{\pi /2}{\int }}\frac{\tan x^{101}}{\tan x^{101}+1}dx$
$=\underset{0}{\overset{\pi /2}{\int }}\frac{1+\tan x^{101}-1}{1+\tan x^{101}}=[x{]}_0^{\pi /2}-I$
$\Rightarrow I=\frac{\pi }{2}-I$
$\Rightarrow 2I=\frac{\pi }{2}$
$\Rightarrow I=\frac{\pi }{4}$