Primitive of Power of Hyperbolic Tangent of a x by Square of Hyperbolic Secant of a x
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Theorem
- $\ds \int \tanh^n a x \sech^2 a x \rd x = \frac {\tanh^{n + 1} a x} {\paren {n + 1} a} + C$
Proof
\(\ds z\) | \(=\) | \(\ds \tanh a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds a \sech^2 a x\) | Derivative of $\tanh a x$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \tanh^n a x \sech^2 a x \rd x\) | \(=\) | \(\ds \int \frac 1 a z^n \rd z\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac {z^{n + 1} } {n + 1}\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\tanh^{n + 1} a x} {\paren {n + 1} a} + C\) | substituting for $z$ and simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tanh a x$: $14.607$