Problem 57 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int x^3 \sqrt{x + c}\ dx $

Answer

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Video Transcript

Let's try to use up here. Let's take you to be explosive so that do you equals DX. Then we also have you might see equals X. So here we see that there's a X cube. So using this equation here, that's you you might see Oops, you might see cubed. And then here, this is just you too. The one half or yeah, You don't have our square, do you? So the next step here, let's just go and evaluate this Cuba. You might seize another hour. So we get you huge minus three you square, See? And then we have plus three you see square and then minus c Cute. So that's the cube right there. And then we're still multiplying by you two don't have power. So before we integrate, let's just go ahead and combined these exponents here. So that's three which is six over too. And then we add the one half and then here have you squared for over two at the one half have the sea there than we have three u to the one which is to over too. So that's you three half C square and then minus. You want have cq. And now let's just go ahead and use the power rule. Not to you, to the nine over to that multiplied by to over nine and then minus. So here we have three C and then you and then we're raising that to the seven over, too. But then we also have to divide by that. So we multiplied by two over seven. Similarly here three c square you to the five half times two over five and then finally here you two, the three half see cubes and then times two or three. And let's go ahead and add that constancy of integration. So just one second and then let's go ahead and just we have two steps last year. One step is to use this equation here to rewrite everything that's from's events. So we'LL go ahead and replace all the use here we have for them, and then we could also headed do some most location here, so three into That's a six and then here you also have a six. So let's just go to the next page to clean this up a little bit. So we'LL have to over mine and then we have X plus C to the nine over, too. And then, after multiplying the two in the three, we had a six and it's policy to the seven halfs. And then we'LL supplying the two of the three again. We're going to six there, five halves and then finally, And let's add our constant of integration Capital seat, which is not the same his three little sea that we see in our final answer. So we'LL start right here, and that's our final answer.