C4 trigonometric integration

there are just some identities for $cos^{2}x$ and $sin^{2}x$ that i've got here on a formula sheet and i've substituted them in, that's all. but i'm sure how to 'recognise' or what to do next, really.

If you have something of the form:

$\displaystyle f'(x)(f(x))^{n}$

Then it's integral dx is $\displaystyle \frac{(f(x))^{n+1}}{n+1} + c$

I should've added a 'k':

$\displaystyle \int kf'(x)(f(x))^n = k\frac{(f(x))^{n+1}}{n+1}$

IN this case, f(x)=cos(4x) and kf'(x) = -1/4 sinx

Sorry I'm in and out and not really paying attention to be frank, but trying to get this stuff back after a long sojourn? Would parts work? Or was I barking up entirely the wrong tree there? I mean on the original equation he gave?

I think so but it requires a lot more work. It is the wrong method to use in this case.

help please!

doing a question now - there's a diagram showing part of a curve $y = 4sinxcos^{3}x$.

I need to find the area enclosed by the curve and the x-axis between x=0 and x=pi. I seem to have lost the plot a bit (excuse me cos I'm ill as well), but I'm confused about what to do!! thanks.

would it be $-4/4cos^{4}x + C$? so then $- cos^{4}x + C$... or have I done that wrong?

I'm also unsure about putting the limits in, would I have to convert it to the actual value and then put them in the calculator in radians?

Yes it is $-cos^4(x)$. I don't know what you mean about the limits. Can you explain further?

I need to find the area enclosed by the curve and the x-axis between x=0 and x=pi.

ohhh i didn't notice that. i assumed the limits were 0 and pi! hopefully if i try that it should work then.

and one LAST question now! - i've got to integrate $cos x (1 + sin^{3}x)$, limits of pi/2 and 0.

how would i go about starting to integrate the above... it seems a bit more complex. even when multiplying it out i end up with $cos x + cos xsin^{3}x$ and that resembles something i've not gone over yet. help appreciated, thank you!

Is there a rule for integrating something like $-1/2 cos2x$?

I'm not too good with latex... the -1/2 means -0.5. and then followed by cos2x.