# Integration By Parts with a definite integral

I've got the following:

\begin{align} \int_{0}^{1}\int_{0}^{y^{2}}\frac{y}{x^{2}+y^{2}}\ dx\ dy&=\int_{0}^{1}\left.\arctan{\left(\frac{x}{y}\right)}\right|_{x=0}^{x=y^{2}}\ dy\\ &=\int_{0}^{1}\arctan{(y)}\ dy\\ &=y\arctan{(y)}-\int_{0}^{1}\frac{y}{1+y^{2}}\ dy \end{align}

I can figure out the integral, but my question is - what should be done about the values of y on the left? I've never done IBP before on a definite integral, so it never came up.

This is supposed to be done without switching the order of integration - I had considered that route but that's not until my next assignment.

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You shouldn't have $y$ left on the first term — it should be $$\begin{bmatrix}y\arctan y\end{bmatrix}^1_0$$
$\dots =y\arctan{(y)}|_{y=0}^{y=1}-\displaystyle\int_{0}^{1}\frac{y}{1+y^{2}}\ dy = \frac{1}{4}\pi -\displaystyle\int_{0}^{1}\frac{y}{1+y^{2}}$