Evaluate each integral.

$$

\int_{0}^{3} y e^{4 x+y^{2}} d y

$$

$\frac{1}{2}\left(e^{4 x+9}-e^{4 x}\right)$

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Johns Hopkins University

Missouri State University

Oregon State University

Harvey Mudd College

{'transcript': "amongst this to take the definite integral from 0 to 3 of wine times the, uh, to the floor ax. Plus. Why square with respect to why? To do this, I'm gonna cover to the Sunday of some use substitution. So you is equal to four x plus. Why square and the derivative of you with respect to why it's just gonna be to I d y. Okay, so in order to put it to in here, I'm gonna multiply the inside, but to the outside about 1/2 okay. To get our do you in there. So now we can write this expression as 1/2 definitely go from 0 to 3 of e to the year. Deal. Okay. And the integral of either the you do you is just eight to the U. So we're gonna evaluate one house times uh, eat of the year, Uh, from 0 to 3. Okay. So, uh, maybe I could do this in, so we're gonna have 1/2 times he to the and then plugging. And three, we're gonna have four x plus nine. Never in its a try and P to the for X plus zero. Okay, So you she might be able to simplify this now. Previously like that to Yeah, that's that's it. So there's your definite integral from 0 to 3 of the infection. Use amuse substitution for it. Thank you very much."}