Primitive of Cosine of a x by Cosine of b x
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Theorem
- $\ds \int \cos a x \cos b x \rd x = \frac {\map \sin {\paren {a - b} x} } {2 \paren {a - b} } + \frac {\map \sin {\paren {a + b} x} } {2 \paren {a + b} } + C$
Proof
| \(\ds \int \cos a x \cos b x \rd x\) | \(=\) | \(\ds \int \paren {\frac {\map \cos {a x - b x} + \map \cos {a x + b x} } 2} \rd x\) | Werner Formula for Cosine by Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \int \map \cos {\paren {a - b} x} \rd x + \frac 1 2 \int \map \cos {\paren {a + b} x} \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \frac {\map \sin {\paren {a - b} x} } {a - b} + \frac 1 2 \frac {\map \sin {\paren {a + b} x} } {a + b} + C\) | Primitive of $\cos a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\map \sin {\paren {a - b} x} } {2 \paren {a - b} } + \frac {\map \sin {\paren {a + b} x} } {2 \paren {a + b} } + C\) | tidying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cos a x$: $14.383$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $62. \ \text {(c)}$