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JH
Numerade Educator

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Problem 8 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int t \sin t \cos t\ dy $

Answer

$\frac{-t \cos (2 t)}{4}+\frac{\sin (2 t)}{8}+C$

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Video Transcript

Let's use integration. My parts for this one. Let's take you to BT So that do you equals DT and then we're left over with Devi equals sign times co sign DT then the You could find this by just integrating. And this in a girl you've seen earlier in the chapter, you could do a use up here or let's use different letter w equal society. And here we'LL get science Where Over too And as usual, when using integration my parts Let's not add the constant C right now. Well, add that in the very end. So here, using the integration by parts you have beauty minus integral video Long has put Are you Indian? So this is our you and Harvey and then minus and a girl. And now we have V again and then do you? So we pull out that to outside the integral in half and then we just have signed square. Now for this you'LL use the half ingle formula If the right is one minus co sign to tea all over too. So let's go ahead and write that. And so combining those twos we get one over four in general one minus cose. Ianto T. Now there's two here. If that's bothering you, you could do another use up here. Let's, um, he's a different different letter here instead of you. Since it's already being used here, you could do another u sub and we should get T Thanks. Word over to still minus. And now we have one over four with the minus, and then that integral of one becomes a T. So that's minus t over. For here. We have a double minus that becomes a plus, and then we have one over four signed t over two plus e. Finally, we got our constant of integration and the last step here isjust combined just a multiply those denominators out. So now we have plus sign to t over eight plus e, and that's our answer.