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I'm trying to solve the following integral:

$$\int\sin^5(x)\cos(x)$$

I assumed I would do u-substitution where:

$$u = \sin(x)$$

$$du = \cos(x) dx$$

Which would then cancel out the $\cos(x)$

And leave me with:

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

But apparently that is not correct?

Update: Seems it is the correct answer. The system I was using gave a different answer, so I plugged in a value into both the system's answer and my own answer, and got different results. Not sure why, but you can consider this closed then.

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    $\begingroup$ $+C$ you forgot $\endgroup$
    – AHusain
    Commented Oct 15, 2018 at 0:21
  • $\begingroup$ What do you mean, "it's not correct?" Did you follow all the instructions your online homework system (coughcough Webassign) gave you? $\endgroup$
    – Sean Roberson
    Commented Oct 15, 2018 at 0:24
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    $\begingroup$ This is fully correct. Actually no substitution is needed, because this is in the form $\int g'(f(x))f'(x)\,dx=g(f(x))+c$, with $g(x)=x^6/6$ and $f(x)=\sin x$. $\endgroup$
    – egreg
    Commented Oct 15, 2018 at 0:24
  • $\begingroup$ @SeanRoberson So the answer they gave is in a different form, and I tried plugging in a value into their answer, and a value into my answer, and I got different results. So I assumed they weren't the same but I guess they are. $\endgroup$
    – James Mitchell
    Commented Oct 15, 2018 at 0:29
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    $\begingroup$ If differentiating the two solutions gives back $\sin^5x\cos x$, then both are correct. $\endgroup$
    – egreg
    Commented Oct 15, 2018 at 0:31

1 Answer 1

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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 $$\int\sin^5(x)\cos(x)=\int\sin(x) (1-\cos^2(x))^2 \cos(x) dx$$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 $$\sin^5(x)\cos(x)=\left(\frac{1-\cos(2x)}{2}\right)^2\frac{\sin(2x)}{2}$$ and then the substitution $u=\cos(2x)$.

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