Problem

Evaluate the following integrals using integratio…

03:39

Problem 24 Medium Difficulty

Evaluate the following integrals using integration by parts.
$$\int x^{2} e^{4 x} d x$$

Answer

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Calculus 1 / AB

Calculus 2 / BC

Calculus Early Transcendentals

Chapter 8

Integration Techniques

Section 2

Integration by Parts

Discussion

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Video Transcript

got integral of X squared times Eat the four x dx So we're gonna use X squared as you and each the four x dx ez devi So you is X squared and do you is two x So Devi is e to the forex. In that case, V is e to the four x over four. All right, now let's we know that integrate by parts, it's integral. A few d v is equal to ah you v minus. And to grow off v do you? So let's start, uh, putting in what we have so x squared times e to the four x over four times integral of v do you? So that's two x times e to the forex over four. So let's cancel out two and four. That becomes too. So we'll take half out. So x squared you to the forex Over four minus God. Put the minus here minus and to go minus half integral of x E to the forex. Well, now we still have X times eat forex. So again, we're gonna do this integral by parts. So again, I'll take you as X and Devi as e to the forex. So Let's ah, derive you. So do use one. Now that we have it as one, it will be easier for us to salt from there. So do you use one V will be Eat the four X over four. So let's move from there. So x e to the forex. Uh, over four minus half into Remember, it's UV minus view. Do you? So you is X each the forex over four minus into go of V, do you? So that's also eat to the four X over for so x e to the forex over four. Remember, this is what we have here. Keep it a squared two X squared e to the forex. Over four, minus half ex. Eat the forex over four minus. So integral feet. The forex over four is each of forex of a force that becomes e to the forex Over 16. Uh, you can distribute half. Uh, that won't be no sorry. Though you might know you can keep the equation as such. Her So add your constant here at the end. So this is your final answer.

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