How do i test out convergence of improper integral
Given integral is $$\int_0^{\infty}\frac{x^m\cos(ax)}{(1+x^n)}dx$$
Answer is the given improper integral is convergent if $-1 < m < n $. I am thinking of applying p test, but how do i deal with $\cos(ax)$ term in numerator?
Thanks
Some hints: The integral may not converge absolutely; the problem is at $\infty.$ Let $m=1,n=2$ to see this. You don't need to know Dirichlet to decide whether the integral is convergent. Try integrating by parts: Integrate $\cos (ax),$ differentiate $x^m/(1+x^n).$