I know how to compute $\displaystyle\int_{0}^\infty \dfrac{\sin x}{x(x^2+1)}dx$. In fact, we can compute $\displaystyle\int_{-\infty}^\infty \dfrac{\sin x}{x(x^2+1)}dx$ and just use the fact that $f(x)=\dfrac{\sin x}{x(x^2+1)}$ is an even function. Now, I have this question and I do not know how to solve it.
Compute $$\displaystyle\int_{0}^\infty \dfrac{\cos x}{x(x^2+1)}dx.$$ Here, we can compute $\displaystyle\int_{-\infty}^\infty \dfrac{\cos x}{x(x^2+1)}dx$, which is $0$, but I do not know how to reduce it to $\displaystyle\int_{0}^\infty \dfrac{\cos x}{x(x^2+1)}dx$. In fact, the function $g(x)=\dfrac{\cos x}{x(x^2+1)}dx$ is odd. Can you give me some hint?
