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When evaluating integrals, especially those involving complex functions like inverse hyperbolic functions, the integration by substitution method often proves helpful. This technique simplifies the integral by substituting a part of the equation with a new variable. It's akin to changing coordinates to ease computations.

To use substitution in the integral \(\int \frac{\sinh^{-1} x}{\sqrt{x^2+1}}\, dx\), we set \(u = \sinh^{-1} x\). This transforms \(x\) into \(\sinh(u)\), which makes finding the differential \( dx \) straightforward. The derivative of \(\sinh(u)\) is \(\cosh(u)\), so \(dx = \cosh(u)\, du\).

Substitution significantly transforms the original integral into a simpler format. The original function's variables and limits get converted using the substitution, affording us an expressed integral, now solely in terms of \(u\). This strategy is advantageous for inverse hyperbolic functions as they have convenient identities and derivatives, allowing for easier integration.

Definite integrals give us the area under the curve of a function, between two specified limits. In essence, they help in calculating the total accumulation of a quantity. The problem provided involves the integral from \(5/12\) to \(3/4\), which defines the bounds of our evaluation.

Through substitution, the original limits, expressed in terms of \(x\), must also be converted to \(u\). Here, \(u = \sinh^{-1}(x)\) yields new limits: the lower limit becomes \(\sinh^{-1}(\frac{5}{12})\) and the upper limit \(\sinh^{-1}(\frac{3}{4})\).

After simplifying the integral through these steps, it is integrated directly using an antiderivative, which when evaluated at these transformed bounds, provides the net area under the function \(\frac{\sinh^{-1} x}{\sqrt{x^2+1}}\) from \(5/12\) to \(3/4\). This definite integral thus gives us a precise numerical result representing the applied scenario.

Hyperbolic identities serve as the backbone for working with hyperbolic functions and their inverses. Similar to trigonometric identities, they allow for the simplification and manipulation of expressions.

In the given problem, we see the use of the identity \(\cosh^2(u) - \sinh^2(u) = 1\). This identity simplifies our work by eliminating the square root, as it converts \(\sqrt{\sinh^2(u) + 1}\) into \(\cosh(u)\).

Recognizing and applying these identities is crucial in the integration process as they often lead to neat simplifications. In this exercise, it enables converting a potentially challenging expression into a conveniently integrable form, aiding in the final computation of the integral.