This is a hard problem to prize in this format. It took me a page on paper. Below are the crucial points:
1) Pull out the 20π. The is simply a constant.
You are watching: Integrate sin^3x cos^2x
2) usage a trig substitution: sin(3x) = 3sin(x) - 4 sin3(x) and also cos(2x) = 1 - 2sin2(x)
3) Multiply out the resulting two binomials to gain 20π∫8sin5(x) -10sin3(x) + 3sin(x) dx
4) rest this into 3 separate integrals. Because that the an initial 160π∫sin5(x), break it into 160π∫sin4(x)sin(x) and then instead of in the trig identity <1-cos2(x)>2 because that sin4(x) then usage a u substitution u = cos(x). The result after you space all perfect is 160π<-cos(x) +2/3cos3(x) - 1/5cos5(x)>
5) because that the 2nd integral -200π∫sin3(x)dx, repeat the process above and you eventually get: -200π<-cos(x) +1/3cos3(x)
6) The 3rd integral easily solves to -60πcos(x)
7) placed them every together and combine prefer terms and you get: -20πcos(x) +40πcos3(x) - 160/5πcos5(x) + C
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Note sin x cos y = 1/2
sin(3x) cos(2x) = 1/2
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QZ P.you re welcome remember to add the + C for the unknown integral
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