Primitive of Exponential of a x by Hyperbolic Sine of b x

Theorem

Exponential Form

$\ds \int e^{a x} \sinh b x \rd x = \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} - \frac {e^{-b x} } {a - b} } + C$

for $a^2 \ne b^2$.

Hyperbolic Form

$\ds \int e^{a x} \sinh b x \rd x = \frac {e^{a x} \paren {a \sinh b x - b \cosh b x} } {a^2 - b^2} + C$

for $a^2 \ne b^2$.

Also see

• Primitive of $e^{a x} \cosh b x$