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## sanchez9457 2 years ago Double Integral:

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1. sanchez9457

$\int\limits_{0}^{1}\int\limits_{0}^{1} xye^xe^y dydx$

2. sanchez9457

i just can't quite figure this one out! Anyone willing to help?

3. kirbykirby

$\int\limits_{0}^{1}\int\limits_{0}^{1}xye^xe^ydydx=\int\limits_{0}^{1}ye^ydy \int\limits_{0}^{1}xe^xdx$ Because the "ye^y dy" part doesn't depend on x, it's like a constant so you can tak it out of the integral. Now you have a multiplication of two integrals that can be done by Integration By Parts... your typical xe^x integral :)

4. kirbykirby

The answer will be 1 btw

5. tkhunny

$$\int y\cdot e^{y}\;dy = \int y\;d\left(e^{y}\right) = y\cdot e^{y} - \int e^{y}\;dy$$

6. kirbykirby

Do you need more help

7. sanchez9457

@kirbykirby i really like that answer as you just answered another of my questions!! But i need some clarification if possible:

8. sanchez9457

Okay so basically you just said that the double integral equals the multiple of the integral of each variable?

9. kirbykirby

Yes, In this case you can do it because the term $ye^ydy$in the integral doesn't depend on x (recall you can just switch the dydx to dxdy and not worry about the bounds of integration since they are constants and the same for both integrals). So since ye^y dy doesn't depend on x, it acts like a constant (with respect to x), so you can just move it out of the integral

10. sanchez9457

AH!!!!!!!!!!

11. sanchez9457

You @kirbykirby are a genius my good sir!

12. kirbykirby

:) no problem!

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