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

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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

$\blacksquare$


Also see


Sources