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# Integration watch

1. OK, I always seem to have difficulty integrating anything like the following, probably because I have never been taught the theory, or anything like that, behind it, so any advise regarding how to systematically integrate anything similar to the following will be greatly appreciated.

A simple example:

Integral sin^4 x dx [thats (sin x)^4]

Or Integral sin^4 x cos^2 x dx [thats (sin x)^4 (cos x)^2]

I believe that you have to switch it using such forumlae as the the double angle formulae, but how do I go about it, seeing that these double angle formulae only contain square terms, not terms in the 4th power (or 6th power, etc).

Any help will be greatly appreciated.
2. INT[sin^4(x)] and so on comes up in P5 (reduction formulae). It is also possible to use complex integration on it, but there's no simple way (someone correct me if I'm wrong) of just integrating something like sin^6(x).cos^4(x).
3. Convert sin^4x into (sin^2x)^2, substitute sin^2x with 1/2(1-sin2x). Very messy business.
4. Anything of the form

(int) sin^(2n)(x)*cos^(2m)(x) dx

can be written as a combination of

(int) sin^2(x) dx,
(int) sin^4(x) dx,
(int) sin^6(x) dx,
etc,

by using sin^2(x) + cos^2(x) = 1. The first of these, (int) sin^2(x) dx, is easy to do. You can use integration by parts to write

(int) sin^4(x) dx in terms of (int) sin^2(x) dx,
(int) sin^6(x) dx in terms of (int) sin^4(x) dx,
(int) sin^8(x) dx in terms of (int) sin^6(x) dx,
etc,

and hence to find (int) sin^4(x) dx, (int) sin^6(x) dx, (int) sin^8(x) dx, etc.

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Updated: June 28, 2004
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