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 Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \int e^{\paren {a + b} x} \rd x - \frac 1 2 \int e^{\paren {a - b} x} \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \frac {e^{\paren {a + b} x} } {a + b} - \frac 1 2 \frac {e^{\paren {a - b} x} } {a - b} + C\) | 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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{a x} \paren {a \sinh b x - b \cosh b x} } {a^2 - b^2} + C\) |
$\blacksquare$