Primitive of Power of Cosine of a x by Sine of a x
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Theorem
- $\ds \int \cos^n a x \sin a x \rd x = \frac {-\cos^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$.
Proof
\(\ds z\) | \(=\) | \(\ds \cos a x\) | Werner Formula for Sine by Cosine | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds -a \sin a x\) | Primitive of $\cos a x$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \cos^n a x \sin a x \rd x\) | \(=\) | \(\ds \int \frac {-z^n \rd x} a\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-z^{n + 1} } {\paren {n + 1} a} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\cos^{n + 1} a x} {\paren {n + 1} a} + C\) | substituting for $z$ |
$\blacksquare$
Also see
For $n = -1$, use Primitive of $\dfrac {\sin a x} {\cos a x}$.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.402$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $66$.