Real Area Hyperbolic Sine of x over a in Logarithm Form

Theorem

\(\ds \arsinh \dfrac x a\)

\(=\)

\(\ds \map \ln {\frac {x + \sqrt {x^2 + a^2} } a}\)

\(\ds \)

\(=\)

\(\ds \map \ln {x + \sqrt {x^2 + a^2} } - \ln a\)

Proof

\(\ds \arsinh \frac x a\)

\(=\)

\(\ds \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 + 1} }\)

Definition of Real Area Hyperbolic Sine

\(\ds \)

\(=\)

\(\ds \map \ln {\frac x a + \sqrt {\frac {x^2 + a^2} {a^2} } }\)

\(\ds \)

\(=\)

\(\ds \map \ln {\frac x a + \frac {\sqrt {x^2 + a^2} } a}\)

\(\ds \)

\(=\)

\(\ds \map \ln {\frac {x + \sqrt {x^2 + a^2} } a}\)

\(\ds \)

\(=\)

\(\ds \map \ln {x + \sqrt {x^2 + a^2} } - \ln a\)

Difference of Logarithms

$\blacksquare$

Also see

• $\arcosh \dfrac x a$ in Logarithm Form

• $\artanh \dfrac x a$ in Logarithm Form

• $\arcoth \dfrac x a$ in Logarithm Form

• $\arsech \dfrac x a$ in Logarithm Form

• $\arcsch \dfrac x a$ in Logarithm Form