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JH
Numerade Educator

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Problem 18 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int \sin x cos \left (\frac{1}{2} x \right) dx $

Answer

$-\frac{4}{3} \cos ^{3}\left(\frac{1}{2} x\right)+C$

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Video Transcript

this problem is from Chapter seven, Section two. Problem number eighteen in the book Capitalist Early Transcendental Sze eighth Edition by James Door And here we have a indefinite noble of sine x times co sign of X over too. So the first thing we can do here is rewrite the sign of eggs. So let's let's write that as sign of two times eggs over too, and then applying the double angle formula over here on the right, we have two sign eggs over too times co signing books over to and we have this extra factory coastline of exit too. So we could simplify this by pulling on it too, combining these co signs. So at this point, we were ready to use u substitution. And here we should take you two be co sign the checks over to. So if we do this, then do you is negative sign except for two. And then using the chain rule left the multiplied by one half and then the ex which we could simplify as negative too. Do you this sign XO over too dx. So if we apply this u substitution, we have a two times negative too So we have a negative for in a girl and then you square, do you, which we can integrate using the pot rule, we get a negative for you, Cube over three plus e and then coming back to our use up. We can replace you with co signing books over to to get our final answer, and that's our answer.