Primitive of Inverse Hyperbolic Cosine of x over a over x squared/Corollary

Theorem

$\ds \int \frac 1 {x^2} \paren {-\cosh^{-1} \dfrac x a} \rd x = -\frac 1 x \paren {-\cosh^{-1} \dfrac x a} - \frac 1 a \arcsec \size {\frac x a} + C$

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

Proof

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

\(=\)

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

Definition of Real Inverse Hyperbolic Cosine

\(\ds \leadsto \ \ \)

\(\ds \int \frac 1 {x^2} \paren {-\cosh^{-1} \dfrac x a} \rd x\)

\(=\)

\(\ds -\int \frac 1 {x^2} \arcosh \dfrac x a \rd x\)

\(\ds \)

\(=\)

\(\ds -\paren {-\frac 1 x \arcosh \dfrac x a + \frac 1 a \arcsec \size {\frac x a} + C}\)

Primitive of $\frac 1 {x^2} \arcosh \dfrac x a$

\(\ds \)

\(=\)

\(\ds -\frac 1 x \paren {-\cosh^{-1} \dfrac x a} - \frac 1 a \arcsec \size {\frac x a} + C\)

Definition of Real Inverse Hyperbolic Cosine

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.655$