SOLUTION: find integrals: sin^3 (x) cos^2(x) . dx the key is ; (1/5) cos^5 (x) -(1/3) cos^3 (x) + c thank

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Question 707038: find integrals: sin^3 (x) cos^2(x) . dx
the key is ; (1/5) cos^5 (x) -(1/3) cos^3 (x) + c
thank
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
int%28sin%5E3%28x%29+cos%5E2%28x%29%2C+dx%29
First factor the sin's:
int%28sin%28x%29sin%5E2%28x%29+cos%5E2%28x%29%2C+dx%29
Then replace sin%5E2%28x%29 with 1-cos%5E2%28x%29:
int%28sin%28x%29%281-cos%5E2%28x%29%29+cos%5E2%28x%29%2C+dx%29
Then multiply:
int%28sin%28x%29%28cos%5E2%28x%29-cos%5E4%28x%29%29%2C+dx%29
int%28sin%28x%29cos%5E2%28x%29-sin%28x%29cos%5E4%28x%29%29%2C+dx%29
Separate into two integrals:
int%28sin%28x%29cos%5E2%28x%29%2Cdx%29-int%28sin%28x%29cos%5E4%28x%29%29%2C+dx%29
Since the derivative of cos is -sin we almost have cos to some power times the derivative of cos in each integral. We just need the factors of -1:

"Pulling" the unwanted -1's outside the integrals:

which simplifies to:

Now we have cos to some power times the derivative of cos. These are now simple integrals:
%28-1%29cos%5E3%28x%29%2F3%2Bcos%5E5%28x%29%2F5+%2B+C
which is equivalent to the answer you provided.