The value of $\int_0^{\frac{\pi}{2}} \frac{\mathrm{d} x}{1+ an ^3 x}$

Question

The value of $\int_0^{\frac{\pi}{2}} \frac{\mathrm{d} x}{1+	an ^3 x}$

Aicharya · Accepted Answer

$\begin{aligned} & \int_0^{\frac{\pi}{2}} \frac{\mathrm{d} x}{1+	an ^3 x}=\int_0^{\frac{\pi}{2}} \frac{\cos ^3 x}{\cos ^3 x+\sin ^3 x} \mathrm{~d} x=\frac{\frac{\pi}{2}-0}{2}=\frac{\pi}{4} \ & {\left[\because \int_a^b \frac{f(x) \mathrm{d} x}{f(x)+f(a+b-x)}=\frac{b-a}{2}ight]}\end{aligned}$

The value of $\int_0^{\frac{\pi}{2}} \frac{\mathrm{d} x}{1+\tan ^3 x}$

Solution