Primitive of x by Square of Hyperbolic Sine of a x
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Theorem
- $\ds \int x \sinh^2 a x \rd x = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C$
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:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \sinh^2 a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {\sinh 2 a x} {4 a} - \frac x 2\) | Primitive of $\sinh^2 a x$ |
Then:
\(\ds \int x \sinh^2 a x \rd x\) | \(=\) | \(\ds x \paren {\frac {\sinh 2 a x} {4 a} - \frac x 2} - \int \paren {\frac {\sinh 2 a x} {4 a} - \frac x 2} \times 1 \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x + \frac 1 2 \int x \rd x + C\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x + \frac {x^2} 4 + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \frac {\cosh 2 a x} {2 a} + \frac {x^2} 4 + C\) | Primitive of $\sinh a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $x \cosh^2 a x$
- Primitive of $x \tanh^2 a x$
- Primitive of $x \coth^2 a x$
- Primitive of $x \sech^2 a x$
- Primitive of $x \csch^2 a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$: $14.548$