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Evaluate the integral.

$ \displaystyle \int x \sin^3 x dx $

$\frac{x \cos ^{3} x}{3}-x \cos x+\frac{2}{3} \sin x+\frac{\sin ^{3} x}{3}+C$

Integration Techniques

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Missouri State University

Harvey Mudd College

University of Nottingham

Boston College

This problem is from Chapter seven section to problem number twenty in the book Calculus Early Transcendental Sze in Tradition by James Door We have an indefinite general of x times signed cume defects. Let's try integration my parts here I think you'd be X. So then d was the ex then we're left over with D is Sign Cube. Becks, the eggs and V is the anti derivative of this, so we will have to compute the integral of Thank you. So let's start up by writing. This is sine squared x Times annex. Then we could apply up the dragon identity to rewrite science where it's co sign squared minus one. Excuse me, We could write sine squared is one minus co sign squared. So again it's from the Pentagon identity. And at this point, we can apply u substitution for this integral. Let's take you to be cosign a Vicks so that negative do you Well, give us the sign. Next dx So separate this from our from our goal over here. So this integral becomes negative in General one minus you swear. Do you evaluating this integral Using the power rule twice We have you minus you cute over three. And there's no need to add the constant of integration here because we're not solving the actual problem that was stated. Right now we're just finding the well at the constant later on at the very end. So here we can distribute the negative, and then we could replace you with co sign from our use up. So we have you cubed over three Cho. Thank you. Over three minus you, which is closer. Okay. So that all this work over here was just to find our key. Now we could use integration my parts on our original problem. So buy in English. My parts This is UV minus and a girl I need you. So we have you Time Z So use X from our own English. My parts Andy was this latest expression down here to an ex Cho Sang cube over three minus x co sign X minus and enroll of again our latest expression v times, do you? Which was just the X. So here we just have a baby. So Cho Sang Kyung, double three minus goes on. So let's simplify this a little bit. Let's come over here to the left Exco Sion cubed over three minus x times co sign We can distribute this negative Signed through on DH Separate this into two winning girls So we get a plus integral of co sign and then we have a minus. Is pulling this wonder in a girl Coach Thank you The VOCs d x So now we have another angle to evaluate. Well, we have two more but we know the integral of course, And assign this one in a world. Cho Sang Qi will take more work so let's separate this one. So for this one, it's actually let's go to the side for this one so cosign cubed So the technique here is going to be very somewhat of the technique that we usedto simplify B So to find V, we needed the integral of sine cuc. Now we have co sign So same idea is going to apply here first seperated and his co sign squared times call sign Then we could rewrite co signed square using the protagonist entity weaken right this guy as one minus dangler Then for this latest in a roll, we can apply any substitution So here we should take unit B sign so that do you is coastline of X, t X. All right, if we do this, integral becomes one minus You squared to you. Which is you? Minus you cubed over three. Yeah. And again we could add to see now. Or we could add it when? Because right now we're not Anak shewell problem that we're working on. We're evaluating, General, of course. Thank you. So let's add the constancy of integration at the very end. At this step, we should come back to our u sub back substitute Sweet sine X minus sank. You'd over three separate that from our scratch work. Okay, so all this was to evaluate intervals co. Thank you. And now we want to come back to our original problem, what we left off. And now we want to plug then the updated information. So our final answer We still have ex coz and cute. So let's maybe try online here. So that's circled. Expression equals ex co sign cubed over three minus x co sign eggs and a girl of coastline is simply sign. Then we have minus one over three times the integral we just evaluated cosign cubed over three so it's going to be negative one over three times the circled expression. So have I. Negative sign eggs over three. The double negative here that's a plus. Signed cute and then three. But we also have a wonder on the bottom. So that's a nine on the bottom plus e. It's our final step here will be to simplify as much as we can. So let's rewrite the last the previous expression that which has had we left off with. Actually, let's go ahead and simplify the sine X minus sign. It's over three, and if we do that, we get a two thirds sign eggs. That's the only simplification necessary, and that's a final answer.