# integral from 0 to infinity of limiting function

If

$$f(x) = \displaystyle\lim_{n\to\infty}{\cos(x)\over1+(\arctan(x))^n},$$ find integral the $\int_0^\infty f(x)\,dx$.

I tried to put $\arctan(x)=t$ and transformed limits to 0 to $\pi/2$ and numerator became derivative of $\sin(\tan(t))$. Then I applied integration by parts but that leads me nowhere.Please help.

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Very bad idea. You can't exchange freely lim and integral (without uniform convergence). Start calculating $$\lim_{n\to->\infty}{\cos(x)\over1+(\arctan(x))^n}.$$