Solving $\iint _D \frac{y}{x^5 + 1} dA$

Question

Solving $$\iint _D \frac{y}{x^5 + 1} dA$$ on $$D = \{(x,y)| 0 \leq x \leq 1, 0 \leq y \leq x^2 \}$$ I have tried to solve it as an iterated integral and I reached this integral which I do not know how to solve $$\frac{1}{6} \int_{0}^{1} \frac{x^2}{x^5 + 1}dx$$

Any help would be appreciated.

Answer 1

Note that\begin{align}\iint_D\frac y{x^5+1}\,\mathrm dx\,\mathrm dy&=\int_0^1\left(\int_0^{x^2}\frac y{x^5+1}\,\mathrm dy\right)\,\mathrm dx\\&=\frac12\int_0^1\frac{x^4}{x^5+1}\,\mathrm dx\\&=\frac1{10}\left[\log(x^5+1)\right]_{x=0}^{x=1}\\&=\frac{\log 2}{10}.\end{align}