Question

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

• A. $\pi /4$
• B. $\pi /3$
• C. 0
• D. None of these

Answer 1

The correct option is A $\pi /4$

$I={\int }_0^{\pi }\frac{1}{1+{\left(\tan x\right)}^{1/4}}dx={\int }_0^{\frac{\pi }{2}}\frac{1}{1+{\left(\tan x\right)}^{1/4}}dx$

$2I={\int }_0^{\frac{\pi }{2}}(\frac{1}{1+{\left(\tan x\right)}^{1/4}}+\frac{{\left(\tan x\right)}^{1/4}}{1+{\left(\tan x\right)}^{1/4}})dx={\int }_0^{\frac{\pi }{2}}\left(\frac{1+{\left(\tan x\right)}^{1/4}}{1+{\left(\tan x\right)}^{1/4}}\right)dx$

$2I={\int }_0^{\frac{\pi }{2}}1dx\Rightarrow 2I=\frac{\pi }{2}$

$\Rightarrow I=\frac{\pi }{4}$