Primitive of x by Inverse Hyperbolic Secant of x over a/Corollary

Theorem

$\ds \int x \paren {-\sech^{-1} \frac x a} \rd x = -\dfrac {x^2} 2 \paren {-\sech^{-1} \frac x a} + \dfrac {a \sqrt {a^2 - x^2} } 2 + C$

where $-\sech^{-1}$ denotes the negative branch of the real inverse hyperbolic secant multifunction.

Proof

\(\ds -\sech^{-1} \frac x a\)

\(=\)

\(\ds -\arsech \frac x a\)

Definition of Real Inverse Hyperbolic Secant

\(\ds \leadsto \ \ \)

\(\ds \int x \paren {-\sech^{-1} \frac x a} \rd x\)

\(=\)

\(\ds -\int x \arsech \frac x a \rd x\)

\(\ds \)

\(=\)

\(\ds -\paren {\dfrac {x^2} 2 \arsech \dfrac x a - \dfrac {a \sqrt {a^2 - x^2} } 2 + C}\)

Primitive of $x \arsech \dfrac x a$

\(\ds \)

\(=\)

\(\ds -\dfrac {x^2} 2 \paren {-\sech^{-1} \frac x a} + \dfrac {a \sqrt {a^2 - x^2} } 2 + C\)

Definition of Real Inverse Hyperbolic Secant

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.667$