convergence of an improper integral, depending of a constant "a"

Question

i need to test the convergence of the following integral:

$$\int_{0}^\infty \frac{x\sin(x)}{x^2+a^2} dx ,a\ge0$$

i managed to show that it converges for a=0, using Dirichlet's criteria. any ideas about $a\gt0$?

thx ;p

Answer 1

You can use here also Dirichlet's criteria! Define $g(x) = \frac{x}{x^2+a^2}$ and note that this function is monotone decreasing for $x > a$ and $\lim_{n \rightarrow \infty} g(x) =0$. Since $|\int_a^b \sin(x) \mathrm{d} x| \leq 1$, we get that $$\int_a^\infty g(x) \sin(x) \mathrm{dx} $$ is convergent. We also have that $g(x) \sin(x)$ is a continuous function, thus it is integrable over compact sets. Therefore the initial integral is convergent.