Primitive of x by Hyperbolic Sine of a x
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Theorem
- $\ds \int x \sinh a x \rd x = \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C$
Proof
With a view to expressing the primitive in the form:
- $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds 1\) | Derivative of Identity Function |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \sinh a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {\cosh a x} a\) | Primitive of $\sinh a x$ |
Then:
\(\ds \int x \sinh a x \rd x\) | \(=\) | \(\ds x \paren {\frac {\cosh a x} a} - \int \paren {\frac {\cosh a x} a} \times 1 \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \cosh a x} a - \frac 1 a \int \cosh a x \rd x + C\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \cosh a x} a - \frac 1 a \paren {\frac {\sinh a x} a} + C\) | Primitive of $\cosh a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C\) | simplification |
$\blacksquare$
Also see
- Primitive of $x \cosh a x$
- Primitive of $x \tanh a x$
- Primitive of $x \coth a x$
- Primitive of $x \sech a x$
- Primitive of $x \csch a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$: $14.541$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $119$.
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(27)$ Integrals Involving $\sinh a x$: $17.27.2.$