# Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Sine

## Theorem

$\displaystyle \int \sin^m a x \cos^n a x \rd x = \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {m + n} } + \frac {m - 1} {m + n} \int \sin^{m - 2} a x \cos^n a x \rd x + C$

## Proof

With a view to expressing the primitive in the form:

$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

 $\displaystyle u$ $=$ $\displaystyle \sin^{m - 1} a x$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle a \paren {m - 1} \sin^{m - 2} a x \cos a x$ Derivative of $\sin a x$, Derivative of Power, Chain Rule for Derivatives

and let:

 $\displaystyle \frac {\d v} {\d x}$ $=$ $\displaystyle \sin a x \cos^n a x$ $\displaystyle \leadsto \ \$ $\displaystyle v$ $=$ $\displaystyle \frac {-\cos^{n + 1} a x} {\paren {n + 1} a}$ Primitive of $\cos^n a x \sin a x$

Then:

 $\displaystyle \int \sin^m a x \cos^n a x \rd x$ $=$ $\displaystyle \int \paren {\sin^{m - 1} a x} \paren {\sin a x \cos^n a x} \rd v$ $\displaystyle$ $=$ $\displaystyle \paren {\sin^{m - 1} a x} \paren {\frac {-\cos^{n + 1} a x} {\paren {n + 1} a} }$ Integration by Parts $\displaystyle$  $\, \displaystyle - \,$ $\displaystyle \int \paren {\frac {-\cos^{n + 1} a x} {\paren {n + 1} a} } \paren {a \paren {m - 1} \sin^{m - 2} a x \cos a x} \rd x + C$ $\displaystyle$ $=$ $\displaystyle \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {n + 1} } + \frac {m - 1} {n + 1} \int \sin^{m - 2} a x \cos^{n + 2} a x \rd x + C$ simplifying $\displaystyle$ $=$ $\displaystyle \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {n + 1} }$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \frac {m - 1} {n + 1} \int \sin^{m - 2} a x \cos^n a x \paren {1 - \sin^2 a x} \rd x + C$ Sum of Squares of Sine and Cosine $\displaystyle$ $=$ $\displaystyle \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {n + 1} } + \frac {m - 1} {n + 1} \int \sin^{m - 2} a x \cos^n a x \rd x$ Linear Combination of Integrals $\displaystyle$  $\, \displaystyle - \,$ $\displaystyle \frac {m - 1} {n + 1} \int \sin^m a x \cos^n a x \rd x + C$

Hence after rearranging:

 $\displaystyle$  $\displaystyle \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {n + 1} } + \frac {m - 1} {n + 1} \int \sin^{m - 2} a x \cos^n a x \rd x$ $\displaystyle$ $=$ $\displaystyle \int \sin^m a x \cos^n a x \rd x + \frac {m - 1} {n + 1} \int \sin^m a x \cos^n a x \rd x + C$ $\displaystyle$ $=$ $\displaystyle \frac {n + 1} {n + 1} \int \sin^m a x \cos^n a x \rd x + \frac {m - 1} {n + 1} \int \sin^m a x \cos^n a x \rd x + C$ common denominator $\displaystyle$ $=$ $\displaystyle \frac {m + n} {n + 1} \int \sin^m a x \cos^n a x \rd x + C$ simplifying $\displaystyle \leadsto \ \$ $\displaystyle \int \sin^m a x \cos^n a x \rd x$ $=$ $\displaystyle \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {m + n} } + \frac {m - 1} {m + n} \int \sin^{m - 2} a x \cos^n a x \rd x + C$ simplifying

$\blacksquare$