Question

Solve the equation : ${\int }_0^{\frac{\pi }{2}}\frac{1}{1+\sqrt[4]{4\tan x}}dx=?$

• A. $\pi /4$
• B. $\pi /2$
• C. $0$
• D. $1/3$

Answer 1

Answer: (A)

$\pi /4$

$I={\int }_0^{\pi /2}\frac{dx}{1+\sqrt[4]{4tanx}}$

$={\int }_0^{\pi /2}\frac{dx}{1+\sqrt[4]{4cotx}}$

$={\int }_0^{\pi /2}\frac{\sqrt[4]{4tanx}dx}{1+\sqrt[4]{4tanx}}$

$\Rightarrow 2I={\int }_0^{\pi /2}\frac{1+\sqrt[4]{4tanx}}{1+\sqrt[4]{4tanx}}dx$

$\Rightarrow 2I=\pi /2$

$\Rightarrow I=\pi /4$ [A]