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Problem 19 Medium Difficulty

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \sin^2 x \cos x \ln (\sin x)\ dx $

Answer

$\frac{1}{9} \sin ^{3} x[3 \ln (\sin x)-1]+C$

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Video Transcript

All right, This question wants us to find the value of this integral. So to do this, we need to get it into one of the forms that our table needs. So we need to find a choice of you. So since we see a sign of ex inside of that natural log, that's probably a good guess. So look you equal to sign X. So that means set d'you is equal to co sign x d. X on we see, we do have a co sign there. Indeed. So are integral. Now becomes Thean Tha Gral of well, co sign next D X becomes d'you and then sign squared in front becomes you squared times the natural log of you, Do you now? This is something that are integral. Table can tell us what the value is. So it says, I'll just put this over here. The integral of you to the end. Ellen of you. Do you based on our formula is U to the n plus one over n plus one squared times n plus one Ellen of you minus one plus c so we can see that our inter girls just the n equals two case of this. So this is equal to you to the n plus one, which is three over and plus one squared times the natural log of n plus one times natural log of you minus one plus c. So then, if we look here, we get you cute. Over nine times three Ellen of you minus one. And now all we have to do is plug in. Are you value of sine X? So we get signed cube decks over nine times three natural log sine X minus one plus c and that's our final answer.

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