Primitive of Square of Hyperbolic Secant of a x over Hyperbolic Tangent of a x
Jump to navigation
Jump to search
Theorem
- $\ds \int \frac {\sech^2 a x \rd x} {\tanh a x} = \frac 1 a \ln \size {\tanh a x} + C$
Proof
\(\ds \frac {\d} {\d x} \tanh a x\) | \(=\) | \(\ds a \sech^2 a x\) | Derivative of $\tanh a x$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {a \sech^2 a x \rd x} {\tanh a x}\) | \(=\) | \(\ds \ln \size {\tanh a x} + C\) | Primitive of Function under its Derivative | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\sech^2 a x \rd x} {\tanh a x}\) | \(=\) | \(\ds \frac 1 a \ln \size {\tanh a x} + C\) | Primitive of Constant Multiple of Function |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tanh a x$: $14.608$