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What is the integral of #int sin^3(x) cos^2(x) dx#?

Answer 1

Samantha Adkison

# (cos^5 x )/5 - (cos^3 )/3 +C #

Write #int sin^3 cos^2 x dx= int sinx (1-cos^2 x) cos^2 x dx#

Now let cos x =t, so that sin dx = -dt. On this substitution, the integral becomes

# -int (1-t^2)(t^2)dt #

#int (t^4-t^2 )dt# = # t^5 /5 -t^3 /3 +C#

No back substitution would have # (cos^5 x )/5 - (cos^3 )/3 +C #

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Answer 2

Adam Adkins

The integral of sin^3(x) * cos^2(x) dx is:

(1/5) * sin^5(x) + C

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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