Question

$\text{Statement-I}:$ The value of the integral $\underset{\pi /6}{\overset{\pi /3}{\int }}\frac{dx}{1+\sqrt{\tan x}}$ is equal to $\frac{\pi }{6}$.
$\text{Statement-II}:\underset{a}{\overset{b}{\int }}f\left(x\right)d\left(x\right)=\underset{a}{\overset{b}{\int }}f(a+b-x)d\left(x\right)$.

• A. Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
• B. Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.
• C. Statement-I is true; Statement-II is false.
• D. Statement-I is false; Statement-II is true.

Answer 1

The correct option is D Statement-I is false; Statement-II is true.

$I=\underset{\pi /6}{\overset{\pi /3}{\int }}\frac{dx}{1+\sqrt{\tan x}}...\left(1\right)$
$=\underset{\pi /6}{\overset{\pi /3}{\int }}\frac{dx}{1+\sqrt{\tan (\frac{\pi }{3}+\frac{\pi }{6}-x)}}$
$=\underset{\pi /6}{\overset{\pi /3}{\int }}\frac{dx}{1+\sqrt{\cot x}}$
$=\underset{\pi /6}{\overset{\pi /3}{\int }}\frac{\sqrt{\tan x}}{1+\sqrt{\tan x}}dx...\left(2\right)$
Adding $\left(1\right)$ and $\left(2\right)$
$\Rightarrow 2I=\underset{\pi /6}{\overset{\pi /3}{\int }}\frac{1+\sqrt{\tan x}}{1+\sqrt{\tan x}}dx$
$\Rightarrow 2I=\underset{\pi /6}{\overset{\pi /3}{\int }}1dx$
$\Rightarrow 2I=\frac{\pi }{6}$
$\Rightarrow I=\frac{\pi }{12}$
$\therefore$ Statement-I is not true and Statement-II is the property
So, Statement-II is always true