The function $\sinh(x)$ is called the hyperbolic sine. It is similar to the regular sine function in trigonometry but is based on hyperbolas rather than circles.

Definition of Hyperbolic Sine:

The hyperbolic sine of $x$ is defined as:

$\sinh(x) = \frac{e^x - e^{-x}}{2}$

Here, $e^x$ is the exponential function.

Inverse Hyperbolic Sine (arsinh):

The inverse hyperbolic sine, denoted $\sinh^{-1}(x)$ or $\text{arsinh}(x)$, is the function that "undoes" the hyperbolic sine. It is defined as:

$\sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1})$

So, if you have $y = \sinh^{-1}(x)$, then: $\sinh(y) = x$

How does $\sinh(x)$ relate to regular sine?

• While $\sin(x)$ (the sine function) is related to the geometry of circles, $\sinh(x)$ is related to the geometry of hyperbolas.
• Hyperbolic functions like $\sinh(x)$ arise in various areas, especially in physics, engineering, and calculus involving certain integrals (like the one you asked about).

Why did $\sinh^{-1}(x)$ appear in the integral?

In integrals involving square roots of sums of squares, the inverse hyperbolic sine function often appears due to its connection with the form of the integrand. Specifically, integrals of the type:

[ \int \frac{du}{\sqrt{u^