Primitive of Hyperbolic Sine of a x by Cosine of p x

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \sinh a x \cos p x \rd x = \frac {a \cosh a x \cos p x + p \sinh a x \sin p x} {a^2 + p^2} + C$


Proof

\(\ds \int \sinh a x \cos p x \rd x\) \(=\) \(\ds \int \paren {\frac {e^{a x} - e^{- a x} } 2} \cos p x \rd x\) Definition of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \frac 1 2 \int e^{a x} \cos p x \rd x - \frac 1 2 \int e^{- a x} \cos p x \rd x\) 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
\(\ds \) \(=\) \(\ds \frac 1 {\paren {a^2 + p^2} } \paren {a \cosh a x \cos p x + p \paren {\frac {e^{a x} - e^{-a x} } 2} \sin p x} + C\) Definition of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \frac 1 {\paren {a^2 + p^2} } \paren {a \cosh a x \cos p x + p \sinh a x \sin p x} + C\) Definition of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \frac {a \cosh a x \cos p x + p \sinh a x \sin p x} {a^2 + p^2} + C\) simplifying

$\blacksquare$


Also see


Sources