Question

The value of the definite integral ${\int }_0^{\infty }\frac{dx}{(1+x^a)(1+x^2)}(a>0)$ is

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

Answer 1

The correct option is A $\frac{\pi }{4}$

$I={\int }_0^{\infty }\frac{dx}{(1+x^a)(1+x^2)}$
Let $x=\tan \theta$
$dx={\sec }^2\theta d\theta$
$I={\int }_0^{\frac{\pi }{2}}\frac{{\sec }^2\theta d\theta }{(1+{\tan }^a\theta )(1+{\tan }^2\theta )}={\int }_0^{\frac{\pi }{2}}\frac{d\theta }{(1+{\tan }^a\theta )}={\int }_0^{\frac{\pi }{2}}\frac{d\theta }{(1+{\tan }^a(\frac{\pi }{2}-\theta ))}={}^{}{\int }_0^{\frac{\pi }{2}}\frac{d\theta }{(1+{\cot }^a\theta )}={\int }_0^{\frac{\pi }{2}}\frac{{\tan }^a\theta d\theta }{(1+{\tan }^a\theta )}$
$\Rightarrow I={\int }_0^{\frac{\pi }{2}}d\theta -{\int }_0^{\frac{\pi }{2}}\frac{d\theta }{(1+{\tan }^a\theta )}=\frac{\pi }{2}-I$
$\Rightarrow I=\frac{\pi }{4}$
Ans: A