$\int\limits_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}$ is :
Answer & explanation
Correct answer: option 4
$I=\int\limits_0^{\pi / 2} \frac{\cos ^3 x}{\sin ^3 x+\cos ^3 x} d x$
$I=\int\limits_0^{\pi / 2} \frac{\sin ^3 x}{\sin ^3 x+\cos ^3 x} d x \Rightarrow 2 I=\int\limits_0^{\pi / 2} dx \Rightarrow I=\frac{\pi}{4}$
Hence (4) is the correct answer.