Question

a nice integral

Answer 1

\[\int\limits_{0}^{\infty} \ \frac{dx}{(1+x^2)(1+\sqrt[n]{x})} \ \ \ \ \ \ \ \]

Answer 2

Is n any positive integer?

Answer 3

n>0 is a real number

Answer 4

\[x=\tan \theta \ \ \ dx=(1+\tan^2{\theta})d \theta \ \ \ \ \theta \ from \ 0 \to \ \frac{\pi}{2} \\ then \ \ \ I=\int\limits_{0}^{\pi/2} \ \frac{d \theta}{1+\sqrt[n]{\tan \theta}}=I=\int\limits_{0}^{\pi/2} \ \frac{\sqrt[n]{\cos \theta}}{\sqrt[n]{\sin \theta}+\sqrt[n]{\cos \theta}} d \theta \]

Answer 5

now what

Answer 6

replace theta wid (pi/2-theta)....

Answer 7

exactly

Answer 8

then add both integrals....final anwer pi/4

Answer 9

the thing i did is i subed x = 1/t.....then dx = -1/t^2