Evaluate $\int_{0}^{\pi}\Big\{\int_{y^2}^{\pi}\frac{y\sin x}{x} dx\Big\}dy$

Question

I want to evaluate the following problem. Can anyone help me? Thanks to you in advance. $$\int_{0}^{\pi}\left\{\int_{y^2}^{\pi}\frac{y\sin x}{x} dx\right\}dy$$

Answer 1

Although there is a singularity about $x=0$, one can work with \begin{align*} \lim_{N\rightarrow\infty}\int_{1/N}^{\pi}\int_{y^{2}}^{\pi}\dfrac{y\sin x}{x}dxdy&=\lim_{N\rightarrow\infty}\int_{1/N^{2}}^{\pi}\int_{1/N}^{\sqrt{x}}\dfrac{y\sin x}{x}dydx. \end{align*}

As pointed by @user539887, there is another term, as written in the comment box.

Answer 2

Straightforward using the Taylor series,

$$I=\frac{1}{2}\left(1-\cos(\pi^2)+\pi^2 \operatorname{Si}(\pi)-\pi^2\operatorname{Si}(\pi^2)\right),$$ where $\operatorname{Si}$ is the Sine integral.

Info about Sine integral: http://mathworld.wolfram.com/SineIntegral.html