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Evaluate the given double integrals.$$\int_{0}^{\ln 3} \int_{0}^{x} e^{2 x+3 y} d y d x$$

$\frac{74}{5}$

Calculus 3

Chapter 29

Partial Derivatives and Double Integrals

Section 4

Double Integrals

Partial Derivatives

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we want to evaluate the double integral. That is the integral from zero to L N three of the integral from zero to X. Even the two X plus through uh dy dx This question is challenging our understanding of integration from the standpoint of Multivariate calculus, we're going to solve in two steps. So age step we use one by one or rather single variable integration techniques where in step one we use we conduct the inner integral which is here the integral zero to active duty to expose three. Why do I once we take the result from this, we plug it into the instagram of our outer integral to solve. So we're going to see this on. Our double integral is equal to the integral from zero to Ellen three of one third of the two X plus three wives and zero xd x. Or one third time interval from zero to Ellen three. Either the five x minus the two x dx. Now we can solve this outer integral. To finalize our solution, we have won 15 out of the five X minus one. Either two X from zero to Ellen three. Plugging in our bounds, we obtain final solution 74 divided by five.

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