- #1
DaleSwanson
- 352
- 2
When I did this problem I overlooked the fact that a u-sub of u=cos x would work fine. I ended up using the more complicated trig integrals rules and did this:
∫cos5 x sin x dx
∫(cos2 x)2 cos x sin x dx
∫(1 - sin2 x)2 cos x sin x dx
∫(sin5 x - 2 sin3 + sin x) cos x dx
with u = sin x, du = cos x dx this is a simple integral. However it gives me:
[itex]\frac{1}{6} sin^{6} x - \frac{1}{2} sin^4 x \frac{1}{2} sin^2 x[/itex]
The correct answer is:
[itex]\frac{-cos^{6} x}{6}[/itex]
Wolfram Alpha confirms that the final form of the integral I gave above still gives the correct answer. It also confirms that my answer isn't numerically equivalent to the correct answer. Also that my integration of the u-subed version is correct. Therefore, I must assume there was some mistake in the u-sub itself, either in the initial sub or the replacement after the integration. I cannot see where this mistake is though.
Just to be clear, I get the correct answer when I use the simpler u-sub of u = cos x, and I understand that is the preferred way to go about this problem. However, as far as I can tell the more complicated way I did it should have still produced a correct answer.
∫cos5 x sin x dx
∫(cos2 x)2 cos x sin x dx
∫(1 - sin2 x)2 cos x sin x dx
∫(sin5 x - 2 sin3 + sin x) cos x dx
with u = sin x, du = cos x dx this is a simple integral. However it gives me:
[itex]\frac{1}{6} sin^{6} x - \frac{1}{2} sin^4 x \frac{1}{2} sin^2 x[/itex]
The correct answer is:
[itex]\frac{-cos^{6} x}{6}[/itex]
Wolfram Alpha confirms that the final form of the integral I gave above still gives the correct answer. It also confirms that my answer isn't numerically equivalent to the correct answer. Also that my integration of the u-subed version is correct. Therefore, I must assume there was some mistake in the u-sub itself, either in the initial sub or the replacement after the integration. I cannot see where this mistake is though.
Just to be clear, I get the correct answer when I use the simpler u-sub of u = cos x, and I understand that is the preferred way to go about this problem. However, as far as I can tell the more complicated way I did it should have still produced a correct answer.