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

Theorem

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

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

Proof

\(\ds \int e^{a x} \sinh b x \rd x\)

\(=\)

\(\ds \frac {e^{a x} } 2 \paren {\frac {e^{b x} } {a + b} - \frac {e^{-b x} } {a - b} } + C\)

Primitive of $e^{a x} \sinh b x$: Exponential Form

\(\ds \)

\(=\)

\(\ds \frac {e^{a x} } 2 \paren {\frac {e^{b x} \paren {a - b} } {\paren {a + b} \paren {a - b} } - \frac {e^{-b x} \paren {a + b} } {\paren {a - b} \paren {a + b} } } + C\)

\(\ds \)

\(=\)

\(\ds \frac {e^{a x} } 2 \paren {\frac {e^{b x} \paren {a - b} - e^{-b x} \paren {a + b} } {\paren {a + b} \paren {a - b} } } + C\)

\(\ds \)

\(=\)

\(\ds \frac {e^{a x} } 2 \paren {\frac {e^{b x} \paren {a - b} - e^{-b x} \paren {a + b} } {a^2 - b^2} } + C\)

Difference of Two Squares

\(\ds \)

\(=\)

\(\ds \frac {e^{a x} } {a^2 - b^2} \paren {\frac {a e^{b x} - b e^{b x} - a e^{-b x} - b e^{-b x} } 2} + C\)

\(\ds \)

\(=\)

\(\ds \frac {e^{a x} } {a^2 - b^2} \paren {a \frac {e^{b x} - e^{-b x} } 2 - b \frac {e^{b x} + e^{-b x} } 2} + C\)

\(\ds \)

\(=\)

\(\ds \frac {e^{a x} } {a^2 - b^2} \paren {a \frac {e^b x - e^{-b} x} 2 - b \cosh b x} + C\)

Definition of Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \frac {e^{a x} } {a^2 - b^2} \paren {a \sinh b x - b \cosh b x} + C\)

Definition of Hyperbolic Sine

\(\ds \)

\(=\)

\(\ds \frac {e^{a x} \paren {a \sinh b x - b \cosh b x} } {a^2 - b^2} + C\)

$\blacksquare$

Also see

• Primitive of $e^{a x} \cosh b x$