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Q.
The value of the definite integral $\int\limits_0^{\infty} \frac{d x}{\left(1+x^a\right)\left(1+x^2\right)}(a>0)$ is
Integrals
Solution:
put $x=\tan \theta$
$I =\int\limits_0^{\pi / 2} \frac{ d \theta}{1+(\tan \theta)^{ a }}=\int\limits_0^{\pi / 2} \frac{(\cos \theta)^{ a }}{(\sin \theta)^{ a }+(\cos \theta)^{ a }} d \theta \Rightarrow I =\frac{\pi}{4}$