Expert Answer to "Integral of sin5(x)cos⁡(x)I assumed I would do u-substitution..."

Malachi Novak

Answered question

2022-04-22

Integral of

\sin^{5} (x) \cos (x)

I assumed I would do u-substitution where:

u = \sin (x)

d u = \cos (x) d x

Which would then cancel out the

\cos (x)

And leave me with:

\int u^{5} d u = \frac{u^{6}}{6} + C = \frac{\sin^{6} (x)}{6} + C

Payton Cantrell

Beginner2022-04-23Added 15 answers

Your answer is correct.
Note that by using a different integration method you can get an answer which looks different but it is not.
For example

\int \sin^{5} (x) \cos (x) = \int \sin (x) {(1 - \cos^{2} (x))}^{2} \cos (x) d x

can becalculayted using the substitution

v = \cos (x)

. If you do this, the answer loos different, but that's just an illusion.
Same way, you can use

\sin^{5} (x) \cos (x) = {(\frac{1 - \cos (2 x)}{2})}^{2} \frac{\sin (2 x)}{2}

and then the substitution

u = \cos 2 x

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