Primitive of Hyperbolic Cosine of a x by Cosine of p x
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Theorem
- $\ds \int \cosh a x \cos p x \rd x = \frac {a \sinh a x \cos p x + p \cosh a x \sin p x} {a^2 + p^2} + C$
Proof
\(\ds \int \cosh a x \cos p x \rd x\) | \(=\) | \(\ds \int \paren {\frac {e^{a x} + e^{- a x} } 2} \cos p x \rd x\) | Definition of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int e^{a x} \cos p x \rd x + \frac 1 2 \int e^{- a x} \cos p x \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac {e^{a x} \paren {a \cos p x + p \sin p x} } {a^2 + p^2} } + \frac 1 2 \paren {\frac {e^{-a x} \paren {-a \cos p x + p \sin p x} } {a^2 + p^2} } + C\) | Primitive of $e^{a x} \cos b x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\paren {a^2 + p^2} } \paren {a \paren {\frac {e^{a x} - e^{-a x} } 2} \cos p x + p \paren {\frac {e^{a x} + e^{-a x} } 2} \sin p x} + C\) | factoring | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\paren {a^2 + p^2} } \paren {a \sinh a x \cos p x + p \paren {\frac {e^{a x} + e^{-a x} } 2} \sin p x} + C\) | Definition of Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\paren {a^2 + p^2} } \paren {a \sinh a x \cos p x + p \cosh a x \sin p x} + C\) | Definition of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {a \sinh a x \cos p x + p \cosh a x \sin p x} {a^2 + p^2} + C\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cosh a x$: $14.574$