# How do I calculate $\int_2^5 \int_0^3 \frac{y\epsilon^{-y}}{x+1}\,dx \,dy$?

How do I calculate the double integral of a fraction?

Considering this double integral:

$$\int_2^5 \int_0^3 \frac{y\varepsilon^{-y}}{x+1} \, dx \, dy$$

I tried to begin with: $$\int_0^3 \frac{y\varepsilon^{-y}}{x+1} \, dx = \left. \frac{1}{x+1} \right|_0^3$$ considering that I was solving for $x$ and the numerator only has $y$ in it, but I don't really know why the denominator is not considered.

My result for this first integral was $-\frac{3}{4}$, but I'm not sure about it and don't know how to continue for the second integral.

## 1 Answer

The integrand is a product of independent functions, this is the dream case scenario, all you have to do is compute separately $\int_2^5 y e^{-y} dy$ and $\int_0^3 \frac{dx}{x+1}$, and multiply them.

This is because when you integrate with respect to $x$ first (say), which the Fubini theorem allows you to do, everything that depends on $y$ is treated as a multiplicative constant that gets out of the integral.

Then you integrate with respect to $y$ and it is now the result of the first integration that is treated as a constant.