How to convert $ \int_0^1\int_0^\frac{x}{2} \frac{x}{x^2 + y^2}\,dy\,dx $ to polar coordinates?

Question

I study on this integral

$$ \int_0^1\int_0^\frac{x}{2} \frac{x}{x^2 + y^2}\,dy\,dx $$

to convert it to polar coordinates. But I am not unable to work out.

I convert the integral to polar coordinates as

$$ \int_0^{22.5°}\int_0^\frac{1}{\cos(\theta)} \frac{r\cos(\theta)}{r^2}r\,dr\,d\theta $$ $$ = \int_0^{22.5°}\int_0^\frac{1}{\cos(\theta)} \cos(\theta)\,dr\,d\theta $$

But the results of these integrals are not equal. How does that work ?

Thanks in advance.

Answer 1

Notice the upper bound for $\theta$ is not correct.

The area is bounded as a triangle, and the upper side is $y=\frac{1}{2}x$, which corresponds to $\theta= \arctan \frac{1}{2}$, which is not half of $45$ degree.

Others looks good to me.