Primitive of Power of Sine of a x by Cosine of a x
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Theorem
- $\ds \int \sin^n a x \cos a x \rd x = \frac {\sin^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$.
Proof
\(\ds z\) | \(=\) | \(\ds \sin a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds a \cos a x\) | Primitive of $\sin a x$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \sin^n a x \cos a x \rd x\) | \(=\) | \(\ds \int \frac {z^n \rd x} a\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {z^{n + 1} } {\paren {n + 1} a} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin^{n + 1} a x} {\paren {n + 1} a} + C\) | substituting for $z$ |
$\blacksquare$
Also see
- For $n = -1$, use Primitive of $\dfrac {\cos a x} {\sin a x}$.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.401$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $64$.