# Evaluating the Integral ∫∫xexy dxdy from 0≤x≤1, 0≤y≤1

• says
In this case, it would be easiest to integrate with respect to x first. Then use integration by parts to solve the resulting integral. After evaluating, you should get an answer of e-2. In summary, when evaluating the integral ∫∫xexy dxdy with limits 0≤x≤1 and 0≤y≤1, using integration by parts and solving with respect to x first, the answer is e-2. The two statements (ey(y-1)+1)/y2 and (ey - (ey (y-1))/y) - 1/y are equivalent.

## Homework Statement

∫∫xexy dxdy where upper and lower limits are 0≤x≤1, 0≤y≤1
so I complete u substitution and get the integral
1/y2 ∫ u*eu du
Now with integration by parts I end up with
xy*exy-exy/y2
I have to evaluate this integral at 0≤x≤1, 0≤y≤1, as mentioned above.
The problem I have is that after evaluating I get an answer of 1, but I've computed this question into symbolab online and it says the answer is e-2.
I was wondering how these two statements are equivalent (below)
(ey(y-1)+1)/y2

(ey - (ey (y-1))/y) - 1/y

## The Attempt at a Solution

You do not need to use variable substitution to solve this integral. Simply set up the problem as an iterated integral. Then figure out which variable is going to be easiest to start with.

## 1. What is the purpose of evaluating the integral?

The purpose of evaluating the integral is to find the area under the curve of a given function within a specific region. It is a way to measure the total value or quantity represented by the function.

## 2. How is the integral evaluated?

The integral is evaluated by using techniques such as substitution, integration by parts, and partial fractions. In this specific example, the integral is evaluated using double integration, which involves integrating with respect to both x and y variables.

## 3. Why are the limits of integration from 0 to 1?

The limits of integration are determined by the specified region, in this case, the square with sides of length 1. The limits of integration define the boundaries within which the function is being evaluated.

## 4. What is the significance of the function exy in this integral?

The function exy is the integrand, which represents the rate of change of the function with respect to both x and y. In this integral, it is being multiplied by the infinitesimal area element dxdy, which allows us to calculate the total value or quantity represented by the function within the given region.

## 5. Can the integral be solved without using double integration?

Yes, the integral can be solved using other methods such as using a computer program or using numerical integration techniques. However, in this specific case, double integration is the most efficient way to evaluate the integral.