Primitive of Hyperbolic Sine of a x over x
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Theorem
\(\ds \int \frac {\sinh a x \rd x} x\) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \frac {\paren {a x}^{2 k + 1} } {\paren {2 k + 1} \paren {2 k + 1}!} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a x + \frac {\paren {a x}^3} {3 \times 3!} + \frac {\paren {a x}^5} {5 \times 5!} + \cdots\) |
Proof
\(\ds \int \frac {\sinh a x \rd x} x\) | \(=\) | \(\ds \int \frac 1 x \paren {\sum_{k \mathop = 0}^\infty \frac {\paren {a x}^{2 k + 1} } {\paren {2 k + 1}!} } \rd x\) | Power Series Expansion for Hyperbolic Sine Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {a^{2 k + 1} } {\paren {2 k + 1}!} \int \frac 1 x \paren {x^{2 k + 1} } \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {a^{2 k + 1} } {\paren {2 k + 1}!} \int x^{2 k} \rd x\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {a^{2 k + 1} } {\paren {2 k + 1}!} \frac {x^{2 k + 1} } {2 k + 1} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \frac {\paren {a x}^{2 k + 1} } {\paren {2 k + 1} \paren {2 k + 1}!} + C\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$: $14.543$