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Use Eq. (5) to evaluate $\int x^{4} e^{x} d x$

$\int x^{4} e^{x} d x=\left[x^{4}-4 x^{3}+12 x^{2}-24 x+24\right] e^{x}+C$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

TECHNIQUES OF INTEGRATION

Section 1

Integration by Parts

Integration

Integration Techniques

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here we will be finding the anti derivative of X to the power four times. Either. The X one strategy that does work is to use integration by parts four times according to this power before it is very tedious to work through all that set up. And that's why we're going to be applying the reduction formula that we see here. The reduction in the reduction formula is that the X to the power of in will give us a new integral, with the power reduced by one. So let's give this a try. So if we use the reduction formula, we write x to the n were in our case and it's four becomes X power four times either the ex minus four times the anti derivative of X to the power of one less so X cubed times either the ex DX. Then at this stage we're at the same conundrum where we could use integration by parts three times, but that's very tedious. Instead, let's associate here to hear and use a reduction formula a second time. Well, together we're really going to apply this reduction formula with X power 44 times. So in the next step. We have X power four times into the X minus four times the result of the reduction formula used here. By that formula, we get altogether X cubed times either the Ex minus three times the anti derivative of X to the power of one less So now it's X squared times either the ex DX. Now let's do a couple operations. At this step, we can distribute to the power of negative four into the group so that we get X power for times either the ex minus four x cubed times of the X Plus 12 when we distribute here out to here times another result that we're going to get by using the reduction formula on this quantity. Here the reduction formula now with and equals two, tells us we get X squared times e to the X minus two times the anti derivative off X to the power of one less. So it's just x this time times either the ex dx Next. If we distribute the 12 through this art, these are the terms were about to pick up. We have expire for either the ex mice four x cubed Either the ex plus 12 x squared either the ex, minus 24 times the anti derivative of X times. Either the ex here we could just use the ordinary integration by parts to solve this. But that's exactly what the reduction for him is giving us. So we may as well reply that formula this time when, and this one and his one will give us just X to the par one or X times, either the ex minus end or just minus the anti derivative off. Now we have here X to the power zero, which is a one. So just either the X DX and the anti driven of either the X is itself. Let's begin to simplify this. We have expire for either the ex minus four x cubed either the ex plus 12 x squared either the ex and now we're going to distribute negative four or excuse me, negative 24. Inside, we get minus 24 x times either the ex plus 24 times anti derivative of either the ex, which is itself and then add a constant of integration. It's fine to leave our answer in this form, but note that if we factor out either the X We can compact this quite a bit. We get x four, then notice we get minus the derivative of X power for which is four x power three. Then we got two plus the derivative of negative four X powers or four X Power three, which is 12 x squared, minus 24 x and plus 24 so on down the line in that way, plus a constant. And this is our full evaluation of that indefinite integral.

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