# Thread: Powers of sines and cosines

1. ## Powers of sines and cosines

Say we want to find $\displaystyle \int\sin^m{x}\cos^n{x}\;{dx}$. Then we would make use of the identity $\displaystyle \cos^2{x}+\sin^2{x}= 1$, if either $\displaystyle m$ or $\displaystyle n$ is odd, and $\displaystyle \cos^2x = \dfrac{1+\cos{2x}}{2}$ and $\displaystyle \sin^2x = \dfrac{1-\cos{2x}}{2}$, if m and n are both even. But isn't this impractical when m and n are large? When want to find, for example, $\displaystyle \int\sin^{78}{x}\cos^{36}{x}\;{dx}$? Or am I missing some trick?

2. Originally Posted by Elvis
Say we want to find $\displaystyle \int\sin^m{x}\cos^n{x}\;{dx}$. Then we would make use of the identity $\displaystyle \cos^2{x}+\sin^2{x}= 1$, if either $\displaystyle m$ or $\displaystyle n$ is odd, and $\displaystyle \cos^2x = \dfrac{1+\cos{2x}}{2}$ and $\displaystyle \sin^2x = \dfrac{1-\cos{2x}}{2}$, if m and n are both even. But isn't this impractical when m and n are large? When want to find, for example, $\displaystyle \int\sin^{78}{x}\cos^{36}{x}\;{dx}$? Or am I missing some trick?
You can create reduction formulae using integration by parts.

These reduction formulae are:

$\displaystyle \int{\sin^m{x}\cos^n{x}\,dx} = \frac{\sin^{m + 1}{x}\cos^{n - 1}{x}}{m + n} + \frac{n - 1}{m + n}\int{\sin^m{x}\cos^{n - 2}{x}\,dx}$

OR

$\displaystyle \int{\sin^m{x}\cos^n{x}\,dx} = -\frac{\sin^{m - 1}{x}\cos^{n + 1}{x}}{m + n} + \frac{m - 1}{m + n}\int{\sin^{m - 2}{x}\cos^n{x}\,dx}$.