Primitive of Exponential of a x by Hyperbolic Cosine of b x/Hyperbolic Form
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Theorem
- $\ds \int e^{a x} \cosh b x \rd x = \frac {e^{a x} \paren {a \cosh b x + b \sinh b x} } {a^2 - b^2} + C$
for $a^2 \ne b^2$.
Proof
\(\ds \int e^{a x} \cosh b x \rd x\) | \(=\) | \(\ds \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} + \frac {e^{-b x} } {a - b} } + C\) | Primitive of $e^{a x} \cosh b x$: Exponential Form | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} } 2 \paren {\frac {e^{b x} \paren {a - b} } {\paren {a + b} \paren {a - b} } + \frac {e^{-b x} \paren {a + b} } {\paren {a - b} \paren {a + b} } } + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} } 2 \paren {\frac {e^{b x} \paren {a - b} + e^{-b x} \paren {a + b} } {\paren {a + b} \paren {a - b} } } + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} } 2 \paren {\frac {e^{b x} \paren {a - b} + e^{-b x} \paren {a + b} } {\paren {a^2 - b^2} } } + C\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} } {a^2 - b^2} \paren {\frac {a e^{b x} - b e^{b x} + a e^{-b x} + b e^{-b x} } 2} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} } {a^2 - b^2} \paren {a \frac {e^{b x} + e^{-b x} } 2 + b \frac {e^{b x} - e^{-b x} } 2} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} } {a^2 - b^2} \paren {a \frac {e^b x + e^{-b} x} 2 + b \sinh b x} + C\) | Definition of Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} } {a^2 - b^2} \paren {a \cosh b x + b \sinh b x } + C\) | Definition of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \paren {a \cosh b x + b \sinh b x} } {a^2 - b^2} + C\) |
$\blacksquare$