Primitive of Cube of Hyperbolic Tangent of a x/Proof 1
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Theorem
- $\ds \int \tanh^3 a x \rd x = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C$
Proof
| \(\ds \int \tanh^3 a x \rd x\) | \(=\) | \(\ds \int \tanh a x \tanh^2 a x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \tanh a x \paren {1 - \sech^2 a x} \rd x\) | Sum of Squares of Hyperbolic Secant and Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \tanh a x \rd x - \int \tanh a x \sech^2 a x \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\ln \size {\cosh a x} } a - \int \tanh a x \sech^2 a x \rd x\) | Primitive of $\tanh a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C\) | Primitive of $\tanh^n a x \sech^2 a x$: $n = 1$ |
$\blacksquare$