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Problem 2 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int \sin^3 \theta \cos^4 \theta d \theta $


$\frac{-\cos ^{5} \theta}{5}+\frac{\cos ^{7} \theta}{7}+C$


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Video Transcript

This problem is from Chapter seven section to problem number two of the book Calculus Early Transcendental Sze eighth Edition by James Store Here we have a triggered a metric integral sign cubed Times Co sign of the fourth. So the first thing we can do is pull out one of the factors of sign here so we can rewrite this integral a sine squared data coastline of the fourth power of data. And then I'll put the other factor of sign over here on the end. It's Ali we did here was rewrite. Sang Cube is science square time sign. The next thing we could do is rewrite the sine squared using a battalion identity so we can write The science square is one minus co sign squared data and everything else remains the same. So it's just copy and paste this remaining coastline to the fourth Scient Ada. Now we see we're ready to apply a u substitution. So here we should apply the U sub u equals co sign of data so that to you becomes negative scientific data. Did Etta notice that there's no negative sign in front of the Sign Data D data and original inaugural So what we'LL do here is multiply this by a negative one so that we get exactly what we want. Signed data debater Good. So re reading are integral using our u substitution. We have a negative one minus you squared. You're the fourth to you. Where the negative sign. It's coming from the negative sign over here. So we proceed by distributing this year the fourth to the one in the minus you square. So we have ah you to the fourth minus. You're the six deal. So before we integrate, let's just distribute this negative sign So we have Ah, in a girl negative you forth. Plus you're the six, do you? So now we apply the power rule twice so we get negative view to the fifth hour over five. Plus, you're the seven power over seven, plus our constant of integration. And finally we've evaluated the integral. But we should get our final answer in terms of the original variable, which was data. So at this step we apply ru substitution to replace you with co signed data. So we have negative coastline. Fifth power of data over five plus co signed seven power over seven plus see And that's our answer