Question

The value of the definite integral $\underset{0}{\overset{\infty }{\int }}\frac{dx}{\left(1+x^a\right)\left(1+x^2\right)}(a>0)$ is

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{2}$
• C. $\pi$
• D. some function of a.

Answer 1

Answer: (A)

put $x=\tan \theta$
$I=\underset{0}{\overset{\pi /2}{\int }}\frac{d\theta }{1+(\tan \theta {)}^a}=\underset{0}{\overset{\pi /2}{\int }}\frac{(\cos \theta {)}^a}{(\sin \theta {)}^a+(\cos \theta {)}^a}d\theta \Rightarrow I=\frac{\pi }{4}$