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

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Theorem

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


Proof

\(\ds \int e^{a x} \sinh b x \rd x\) \(=\) \(\ds \int e^{a x} \paren {\frac {e^{b x} - e^{-b x} } 2} \rd x\) Definition of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \frac 1 2 \int e^{a x} \paren {e^{b x} - e^{-b x} } \rd x\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \frac 1 2 \int \paren {e^{a x} e^{b x} - e^{a x} e^{-b x} } \rd x\)
\(\ds \) \(=\) \(\ds \frac 1 2 \int \paren {e^{a x + b x} - e^{a x - b x} } \rd x\) Product of Powers
\(\ds \) \(=\) \(\ds \frac 1 2 \int e^{a x + b x} \rd x - \frac 1 2 \int e^{a x - b x} \rd x\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds \frac 1 2 \int e^{\paren {a + b} x} \rd x - \frac 1 2 \int e^{\paren {a - b} x} \rd x\)
\(\ds \) \(=\) \(\ds \frac 1 2 \frac {e^{\paren {a + b} x} } {a + b} - \frac 1 2 \frac {e^{\paren {a - b} x} } {a - b} + C\) Primitive of $e^{a x}$
\(\ds \) \(=\) \(\ds \frac 1 2 \frac {e^{a x + b x} } {a + b} - \frac 1 2 \frac {e^{a x - b x} } {a - b} + C\)
\(\ds \) \(=\) \(\ds \frac 1 2 \frac {e^{a x} e^{b x} } {a + b} - \frac 1 2 \frac {e^{a x} e^{-b x} } {a - b} + C\) Product of Powers
\(\ds \) \(=\) \(\ds \frac 1 2 \frac {e^{a x} e^{b x} \paren {a - b} } {\paren {a + b} \paren {a - b} } - \frac 1 2 \frac {e^{a x} e^{-b x} \paren {a + b} } {\paren {a - b} \paren {a + b} } + C\)
\(\ds \) \(=\) \(\ds \frac {e^{a x} e^{b x} \paren {a - b} - e^{a x} e^{-b x} \paren {a + b} } {2 \paren {a + b} \paren {a - b} } + C\)
\(\ds \) \(=\) \(\ds \frac {e^{a x} e^{b x} \paren {a - b} - e^{a x} e^{-b x} \paren {a + b} } {2 \paren {a^2 - b^2} } + C\) Difference of Two Squares
\(\ds \) \(=\) \(\ds \frac {a e^{a x} e^{b x} - b e^{a x} e^{b x} - a e^{a x} e^{-b x} - b e^{a x} e^{-b x} } {2 \paren {a^2 - b^2} } + C\)
\(\ds \) \(=\) \(\ds \frac {e^{a x} } {\paren {a^2 - b^2} } \paren {\frac {a e^{b x} - b e^{b x} - a e^{-b x} - b e^{-b x} } 2} + C\)
\(\ds \) \(=\) \(\ds \frac {e^{a x} } {\paren {a^2 - b^2} } \paren {a \frac {e^{b x} - e^{-b x} } 2 - b \frac {e^{b x} + e^{-b x} } 2} + C\)
\(\ds \) \(=\) \(\ds \frac {e^{a x} } {\paren {a^2 - b^2} } \paren {a \frac {e^b x - e^{-b} x} 2 - b \cosh b x} + C\) Definition of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \frac {e^{a x} } {\paren {a^2 - b^2} } \paren {a \sinh b x - b \cosh b x } + C\) Definition of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \frac {e^{a x} \paren {a \sinh b x - b \cosh b x} } {a^2 - b^2} + C\)

$\blacksquare$


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