Integrate w/ trig substitution: int sin^3(x) cos^5(x)dx

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MarkSA

Junior Member
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Sep 8, 2007
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Hello,

1) Integrate (using trig substitution):
integral of: (sinx)^3(cosx)^5 dx

I broke it into:
integral: (sinx)^3(cosx)(1 - (sinx)^2)^2 dx
Let u = sinx
du = cosxdx
integral of: u^3(1 - u^2)^2 du
becomes integral of: u^3 - 2u^5 + u^7 du
= 1/4 * (sinx)^4 - 1/3 * (sinx)^6 + 1/8 * (sinx)^8 + C

However the 'correct' answer appears to be:
= 1/8 * (cosx)^8 - 1/6 * (cosx)^6 + C

I thought the different answer was just a result of using sinx instead of cosx for u. But if I pick random values for x, the answers do not output the same value.. so I think they must be different. I can't figure out where I went wrong in the problem I worked out above though.. can anyone identify where?
 
Re: Integrate sin^3xcos^5xdx

Picking random values won't work - because there is an "integration constant" - which could be different for different procedures.

Look at the difference of the 'answers' from evaluating those functions. Are those constant?

Differentiate your answer - do you get back the original integrand?
 
Re: Integrate sin^3xcos^5xdx

Thanks, testing this way has worked before but I hadn't thought about the + C being different.

Unfortunately the difference in the answers isn't constant. I guess this means they are definitely different answers?

I've looked over mine again but I still can't see any mistakes in it. it was a fairly short problem
 
Re: Integrate sin^3xcos^5xdx

\(\displaystyle f(x) = \frac{1}{4}sin^{4}x - \frac{1}{3}sin^{6}x + \frac{1}{8}sin^{8}x\)

\(\displaystyle \rightarrow f\left(\frac{\pi}{4}\right) \approx 0.028645833\)

\(\displaystyle \rightarrow f\left(\frac{\pi}{3}\right) \approx 0.0395507813\)


\(\displaystyle g(x) = \frac{1}{8}cos^{8}x - \frac{1}{6}cos^{6}x\)

\(\displaystyle \rightarrow g\left(\frac{\pi}{4}\right) \approx -0.130208333\)

\(\displaystyle \rightarrow g\left(\frac{\pi}{3}\right) \approx -0.0021158854\)

Putting them together ...
\(\displaystyle f\left(\frac{\pi}{4}\right) - g\left(\frac{\pi}{4}\right) = 0.04166666 \: ... \approx \frac{1}{24}\)

\(\displaystyle f\left(\frac{\pi}{3}\right) - g\left(\frac{\pi}{3}\right) = 0.04166666 \: ... \approx \frac{1}{24}\)
 
I checked those at 0 and ? - and those gave constant difference. So the sine function is equivalent to cosine function.
 
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