Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Cosine
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Theorem
- $\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {\cos^{m - 1} a x} {a \paren {m - n} \sin^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\cos^{m - 2} a x} {\sin^n a x} \rd x + C$
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:
\(\ds u\) | \(=\) | \(\ds \frac {\cos^{m - 1} a x} {\sin^n a x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac {\sin^n a x \dfrac \d {\d x} \cos^{m - 1} a x - \cos^{m - 1} a x \dfrac \d {\d x} \sin^n a x} {\sin^{2 n} a x}\) | Quotient Rule for Derivatives |
Thus:
\(\ds \frac \d {\d x} \cos^{m - 1} a x\) | \(=\) | \(\ds a \paren {m - 1} \cos^{m - 2} a x \paren {-\sin a x}\) | Derivative of $\cos a x$, Derivative of Power, Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds -a \paren {m - 1} \cos^{m - 2} a x \sin a x\) | ||||||||||||
\(\ds \frac \d {\d x} \sin^n a x\) | \(=\) | \(\ds a n \sin^{n - 1} a x \cos a x\) | Derivative of $\sin a x$, Derivative of Power, Chain Rule for Derivatives |
and so:
\(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \frac {\sin^n a x \paren {-a \paren {m - 1} \cos^{m - 2} a x \sin a x} - \cos^{m - 1} a x \paren {a n \sin^{n - 1} a x \cos a x} } {\sin^{2 n} a x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-a \paren {m - 1} \cos^{m - 2} a x \sin^2 a x + a n \cos^m a x} {\sin^{n + 1} a x}\) | simplifying and cancelling $\sin^{n - 1}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-a \cos^{m - 2} a x \paren {\paren {m - 1} \sin^2 a x + n \cos^2 a x} } {\sin^{n + 1} a x}\) | factorising | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-a \cos^{m - 2} a x \paren {\paren {m - 1} \paren {1 - \cos^2 a x} + n \cos^2 a x} } {\sin^{n + 1} a x}\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-a \cos^{m - 2} a x \paren {\paren {n - m + 1} \cos^2 a x + \paren {m - 1} } } {\sin^{n + 1} a x}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-a \paren {n - m + 1} \cos^m a x} {\sin^{n + 1} a x} + \frac {a \paren {m - 1} \cos^{m - 2} a x} {\sin^{n + 1} a x}\) | separating |
Then let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \cos a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {\sin a x} a\) | Primitive of $\cos a x$ |
Then:
\(\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x\) | \(=\) | \(\ds \int \frac {\cos^{m - 1} a x} {\sin^n a x} \cos^a x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {\cos^{m - 1} a x} {\sin^n a x} } \paren {\frac {\sin a x} a}\) | Integration by Parts | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \int \paren {\frac {\sin a x} a} \paren {\frac {-a \paren {n - m + 1} \cos^m a x} {\sin^{n + 1} a x} + \frac {a \paren {m - 1} \cos^{m - 2} a x} {\sin^{n + 1} a x} } \rd x + C\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cos^{m - 1} a x} {a \sin^{n - 1} a x}\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {n - m + 1} \int \frac {\cos^m a x} {\sin^n a x} \rd x\) | Linear Combination of Primitives | ||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {m - 1} \int \frac {\cos^{m - 2} a x} {\sin^n a x} \rd x + C\) |
\(\ds \leadsto \ \ \) | \(\ds \paren {1 - \paren {n - m + 1} } \int \frac {\cos^m a x} {\sin^n a x} \rd x\) | \(=\) | \(\ds \frac {\cos^{m - 1} a x} {a \sin^{n - 1} a x} + \paren {m - 1} \int \frac {\cos^{m - 2} a x} {\sin^n a x} \rd x + C\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {m - n} \int \frac {\cos^m a x} {\sin^n a x} \rd x\) | \(=\) | \(\ds \frac {\cos^{m - 1} a x} {a \sin^{n - 1} a x} + \paren {m - 1} \int \frac {\cos^{m - 2} a x} {\sin^n a x} \rd x + C\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x\) | \(=\) | \(\ds \frac {\cos^{m - 1} a x} {a \paren {m - n} \sin^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\cos^{m - 2} a x} {\sin^n a x} \rd x + C\) |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.427$