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

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Theorem

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

for $n \ne -m$.


Proof

With a view to expressing the primitive in the form:

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

let:

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


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \sin^m a x \cos a x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {\sin^{m + 1} a x} {\paren {m + 1} a}\) Primitive of $\sin^n a x \cos a x$


Then:

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


Hence after rearranging:

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

$\blacksquare$


Also see


Sources

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