Question

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

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

Answer 1

The correct option is A $\pi /4$

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

${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx$

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

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

$2I=\frac{\pi }{2}$

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

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