Primitive of Sine of a x by Cosine of b x
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Theorem
- $\ds \int \sin a x \cos b x \rd x = \frac {-\map \cos {a - b} x} {2 \paren {a - b} } - \frac {\map \cos {a + b} x} {2 \paren {a + b} } + C$
for $a, b \in \R: a \ne b$
Proof
\(\ds \int \sin a x \cos b x \rd x\) | \(=\) | \(\ds \int \paren {\dfrac {\map \sin {a x + b x} + \map \sin {a x - b x} } 2} \rd x\) | Werner Formula for Sine by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int \map \sin {a - b} x \rd x + \frac 1 2 \int \map \sin {a + b} x \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \frac {-\map \cos {a - b} x} {a - b} - \frac 1 2 \frac {\map \cos {a + b} x} {a + b} + C\) | Primitive of $\cos a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\map \cos {a - b} x} {2 \paren {a - b} } - \frac {\map \cos {a + b} x} {2 \paren {a + b} } + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $\sin a x \cos a x$ for $a = b$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.400$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $62. \ \text {(a)}$
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(19)$ Integrals Involving $\sin a x$ and $\cos a x$: $17.19.2.$