How to Integrate Even Powers of Sine & Cosine
- 1). Transform the integrand into terms of sine or cosine, using the Pythagorean identity. The Pythagorean identity states that sin^2(x) + cos^2(x) = 1.
If the integrand contains powers of both sine and cosine, then sin^m(x)cos^n(x) can be written as sin^{2k}(x)cos^n(x) = cos^n(x)(1 - cos^2(x))^k. Then the exponentiation can be expanded into a sum of even powers of cosines. For example, if the integrand is sin^4(x)cos^4(x), then the Pythagorean identity yields cos^4(x)(1 - cos^2(x))^2 = cos^4(x)(1 - 2cos^2(x) + cos^4(x)) = cos^4(x) - 2cos^6(x) + cos^8(x). - 2). Use a power reduction integration technique to bring down the power of each term. These identities state that for an even power n, the integral of sin^n(x), Int{sin^n(x)} is:
int{sin^n(x)} = -sin^(n - 1)(x)cos(x)/n + (n - 1) / n * int{sin^(n - 2)(x)}
and for cosine:
int{cos^n(x)} = cos^(n - 1)(x)sin(x) / n+ (n - 1)/n * int{cos^(n - 2)(x)}
Use these identities to bring the powers down to an integral of either cos^2(x) or sin^2(x). In our example above, the integration of the second term of -2cos^6(x) can be reduced to:
int{-2cos^6(x)} = -2[cos^5(x)sin(x) / 6 + 5 / 6 * int{cos^4(x)}] = -2[cos^5(x)sin(x) / 6 + 5 / 6 * (cos^3(x)sin(x) / 4 + 3 / 4 * int{cos^2(x)})] = -cos^5(x)sin(x) / 3 - 5 / 12 * cos^3(x)sin(x) - 15 / 12 * int{cos^2(x)}. - 3). Use the double angle power reduction. The double angle power reduction states that sin^2(x) = (1 - cos(2x)) / 2 and cos^2(x) = (1 + cos(2x)) / 2. These will replace the final cos^2 or sin^2 in the above integration chain. Once substituted in, the integral will be simple to execute:
int{cos^2(x)} = int{(1 + cos(2x))/2} = (2x + sin(2x)) / 4 and int{sin^2(x)} = int{(1 - cos(2x)) / 2} = (2x - sin(2x)) / 4.
In our above example for -2cos^6(x), then
int{-2cos^6(x)} = -cos^5(x)sin(x) / 3 - 5 / 12 * cos^3(x)sin(x) - 15 / 48 * sin(2x) - 15 / 24 * 2x,
with the addition of a constant of integration. The other terms of the original integral can be similarly handled, and the final equation can be reached. - 4). Combine terms and apply limits of integration. At this point, you should have a long string of terms in the integration. Common terms are used to combine various elements of the sum in order to simplify the integration. If there are limits to the integration, then apply them to simplify into a single numerical result. If there are no limits, then leave the equation as is and add a +C in order to consider the constant of integration.
Source...