Primitive of Inverse Hyperbolic Sine Function

Theorem

$\ds \int \arsinh x \rd x = x \arsinh x - \sqrt {x^2 + 1} + C$

Proof

From Primitive of $\arsinh \dfrac x a$:

$\ds \int \arsinh \frac x a \rd x = x \arsinh \frac x a - \sqrt {x^2 + a^2} + C$

The result follows by setting $a = 1$.

$\blacksquare$

Also see

• Primitive of $\arcosh x$
• Primitive of $\artanh x$
• Primitive of $\arcoth x$

Sources

• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals