Primitive of Power of Hyperbolic Sine of a x
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Theorem
- $\ds \int \sinh^n a x \rd x = \frac {\sinh^{n - 1} a x \cosh a x} {a n} - \frac {n - 1} n \int \sinh^{n - 2} a x \rd x$
for $n \ne 0$.
Proof
With a view to expressing the problem 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:
\(\ds u\) | \(=\) | \(\ds \sinh^{n - 1} a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds \paren {n - 1} a \sinh^{n - 2} a x \cosh a x\) | Chain Rule for Derivatives, Derivative of $\sinh a x$, Derivative of Power |
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:
\(\ds \int \sinh^n a x \rd x\) | \(=\) | \(\ds \int \sinh^{n - 1} a x \sinh a x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sinh^{n - 1} a x \paren {\frac {\cosh a x} a}\) | Integration by Parts | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \int \paren {\frac {\cosh a x} a} \paren {\paren {n - 1} a \sinh ^{n - 2} a x \cosh a x} \rd x\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh^{n - 1} a x \cosh a x} a\) | simplifying | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \int \paren {n - 1} \sinh^{n - 2} a x \cosh^2 a x \rd x\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh^{n - 1} a x \cosh a x} a\) | Difference of Squares of $\cosh$ and $\sinh$ | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \int \paren {n - 1} \sinh^{n - 2} a x \paren {1 + \sinh^2 a x} \rd x\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sinh^{n - 1} a x \cosh a x} a - \paren {n - 1} \int \sinh^{n - 2} a x \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {n - 1} \int \sinh^n a x \rd x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds n \int \sinh^n a x \rd x\) | \(=\) | \(\ds \frac {\sinh^{n - 1} a x \cosh a x} a - \paren {n - 1} \int \sinh^{n - 2} a x \rd x\) | rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \sinh^n a x \rd x\) | \(=\) | \(\ds \frac {\sinh^{n - 1} a x \cosh a x} {a n} - \frac {n - 1} n \int \sinh^{n - 2} a x \rd x\) | dividing both sides by $n$ |
We note that if $n = 0$, then $\dfrac {n - 1} n$ is undefined, rendering the derivation invalid.
$\blacksquare$
Also see
- Primitive of $\cosh^n a x$
- Primitive of $\tanh^n a x$
- Primitive of $\coth^n a x$
- Primitive of $\sech^n a x$
- Primitive of $\csch^n a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sinh a x$: $14.558$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $117$.