91Ó°ÊÓ

Inverse hyperbolic functions are counterparts of the regular inverse trigonometric functions, but they are centered around hyperbolic functions such as sinh, cosh, and tanh. One such function is the inverse hyperbolic sine, denoted as \( \sinh^{-1}(x) \) or \( \text{arsinh}(x) \). It reverses the hyperbolic sine function, \( \sinh(x) \), meaning if \( y = \sinh^{-1}(x) \), then \( x = \sinh(y) \).

These functions frequently arise when dealing with integrals involving radicals, such as \( \sqrt{x^2 + 1} \). Inverse hyperbolic functions have similar properties to their trigonometric analogs, but they often provide simpler expressions when solving integrals due to their algebraic characteristics. For example, the derivative of \( \sinh^{-1}(x) \) results in the simpler expression \( \frac{1}{\sqrt{x^2 + 1}} \), which is precisely the structure needed to tackle the integration problem at hand. Thus, recognizing the pattern of these derivatives is crucial in solving integrals involving inverse hyperbolic functions.

An antiderivative, also known as an indefinite integral, represents the reverse process of differentiation. If a function \( f(x) \) has a derivative, then finding its antiderivative means determining a function \( g(x) \) such that \( g'(x) = f(x) \).

In the current exercise, the task is to find the antiderivative of \( f(x) = \frac{x}{\sqrt{x^2 + 1}} \). This involves identifying a function \( g(x) \) where the derivative returns us back to \( f(x) \). By recognizing that \( f(x) \) is linked to the derivative of the inverse hyperbolic funtion \( \sinh^{-1}(x) \), it guides us toward integrating \( f(x) \) to get \( g(x) = \frac{1}{2}\sinh^{-1}(x)^2 + C \), where \( C \) is the constant of integration. This constant represents the infinite number of vertical shifts possible for the family of antiderivatives of \( f(x) \).

Finding antiderivatives can involve various methods such as substitution or partial fraction decomposition, but the underlying goal remains the same: to reverse the action of differentiation.

Integration by substitution is a powerful technique used to simplify integrals by introducing a new variable. It is the reverse application of the chain rule for derivatives.

In this exercise, substitution helps make the integration of \( f(x) = \frac{x}{\sqrt{x^2 + 1}} \) more manageable. By setting \( u = \sinh^{-1}(x) \), we effectively reduce the complexity of the integral because the derivative of \( u \) directly corresponds to components of the integrand. This substitution converts the integral from a more convoluted expression into a simpler form \( \int u du \), which evaluates to \( \frac{1}{2}u^2 + C \).

To apply substitution:

• Identify a substitution that simplifies the integral. Typically, this involves recognizing parts of the integrand as derivatives of another function;
• Express all parts of the integrand and differential in terms of the new variable;
• Solve the simpler integral;
• Re-substitute the original variable expressions to return to the initial framework.

Ultimately, integration by substitution exploits patterns and symmetries within functions to facilitate easier computation of integrals.