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Inverse trigonometric functions are the functions that reverse the action of the regular trigonometric functions. For instance, if the sine function gives us an angle's vertical component when the hypotenuse is 1 (from geometry), the inverse sine function tells us what angle yields that particular sine value.
These functions are particularly important in calculus due to their derivatives, which often appear in integration problems.
Some of the most common inverse trigonometric functions include:

• \ \( \arcsin(x) \ \): Inverse sine function
• \ \( \arccos(x) \ \): Inverse cosine function
• \ \( \arctan(x) \ \): Inverse tangent function
• \ \( \arcsec(x) \ \): Inverse secant function

Each of these functions has distinct derivatives that are used to evaluate integrals, such as the derivative of the inverse secant function, \ \( \frac{d}{dx}(\sec^{-1} x) = \frac{1}{x \sqrt{x^2-1}} \ \). This particular derivative was critical in solving our exercise due to its similarity with the original function integrand.

The domain of a function is the complete set of possible input values (x-values) for which the function is defined.
Understanding the domain is crucial because it tells us where the function exists and behaves according to its equation. A function may not be defined for certain values due to division by zero or taking the square root of a negative number, which makes finding the domain a key step in function analysis.
For the function \ \( f(x) = \frac{1}{x \sqrt{1-x^2}} \ \), several condition affect its domain:

• \ \( x \) cannot be 0 because the function would involve division by 0.
• The term \ \( \sqrt{1-x^2} \) shows the function is undefined for \ \( x = 1 \) and \ \( x = -1 \) because \ \( \sqrt{1-x^2} = 0 \).

Thus, the domain for \ \( f(x) \) is \ (-1, 0) \cup (0, 1) \, representing intervals that avoid these undefined scenarios. Therefore, the domain of its antiderivative \ \( F(x) \) is also the same, as antiderivatives share the domain of their derivative functions.

A table of integrals is a powerful reference tool that contains various integral formulas. These tables are indispensable for solving integrals, particularly when facing complex functions that match specific integral forms found in the table.
Tables typically list standard forms side by side with their corresponding results. Users can identify which form applies to a given problem, possibly adjusting the problem to match one of the listed forms.
Here’s how you might typically use these tables:

• Identify an integral form from the table that matches or is similar to the given integrand.
• Transform the integral, if necessary, using substitution or algebraic manipulations to fit one of the standard forms.
• Use the formula provided in the table for the recognized integral form to quickly find the antiderivative.

In the exercise given, the function \ \( f(x) = \frac{1}{x \sqrt{1-x^2}} \) was identified as a variation of a derivative of an inverse trigonometric function from the table — specifically, related to the derivative of the inverse secant. This connection simplified the integration process by directly applying the tabulated result, demonstrating the practical utility of a table of integrals in calculus.