Derivative of Real Area Hyperbolic Tangent of x over a/Corollary
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Theorem
- $\map {\dfrac \d {\d x} } {\dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } } = \dfrac 1 {a^2 - x^2}$
where $\size x < a$.
Proof
| \(\ds \dfrac 1 a \map {\tanh^{-1} } {\frac x a}\) | \(=\) | \(\ds \dfrac 1 a \cdot \dfrac 1 2 \map \ln {\dfrac {1 + \frac x a} {1 - \frac x a} }\) | Definition of Real Area Hyperbolic Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} }\) | multiplying top and bottom of argument by $a$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map {\dfrac \d {\d x} } {\dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } }\) | \(=\) | \(\ds \map {\dfrac \d {\d x} } {\dfrac 1 a \map {\tanh^{-1} } {\frac x a} }\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 a \cdot \dfrac a {a^2 - x^2}\) | Derivative of Real Area Hyperbolic Tangent of x over a | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {a^2 - x^2}\) | simplifying |
$\blacksquare$
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $16$.