Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 2 down vote accepted

You shouldn't have $y$ left on the first term — it should be $$\begin{bmatrix}y\arctan y\end{bmatrix}^1_0$$

share|cite|improve this answer

The last part should be

$\dots =y\arctan{(y)}|_{y=0}^{y=1}-\displaystyle\int_{0}^{1}\frac{y}{1+y^{2}}\ dy = \frac{1}{4}\pi -\displaystyle\int_{0}^{1}\frac{y}{1+y^{2}}$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.