Problem 56 - Applied Calculus

Question

Find each integral by whatever means are necessary (either substitution or tables). $$ \int \frac{1}{\sqrt{x^{2}-1}} d x $$

Step-by-Step Solution

Answer

\( \ln |x + \sqrt{x^{2}-1}| + C \).

Step 1: Identify the Integral Form

The integral \( \int \frac{1}{\sqrt{x^{2}-1}} \, dx \) resembles a form that is often related to inverse trigonometric functions.

Step 2: Recall Integral Identity

Recall that the integral \( \int \frac{1}{\sqrt{x^2 - a^2}} \, dx = \ln |x + \sqrt{x^2 - a^2}| + C \) for a constant \(a\). In this case, \(a = 1\).

Step 3: Apply the Formula

We directly apply the formula to the given integral with \( a = 1 \), so we get: \( \int \frac{1}{\sqrt{x^{2}-1}} \, dx = \ln |x + \sqrt{x^{2}-1}| + C \).

Step 4: Final Result

The definite answer for the integral is \( \ln |x + \sqrt{x^{2}-1}| + C \), where \(C\) is the constant of integration.

Key Concepts

Inverse Trigonometric Functions

In calculus, inverse trigonometric functions allow us to integrate specific forms into simpler expressions that involve these functions. They can transform complex trigonometric expressions into more manageable formats. Familiar inverse trigonometric functions such as \( \arcsin(x) \), \( \arccos(x) \), and \( \arctan(x) \), correspond to integrals involving expressions under square roots. These functions help to manage integrals of form \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx \), \( \int \frac{1}{\sqrt{x^2 - a^2}} \, dx \), and \( \int \frac{1}{x^2 + a^2} \, dx \) respectively. For our specific integral \( \int \frac{1}{\sqrt{x^{2}-1}} \, dx \), it aligns with the inverse trigonometric form because of the \( x^2 - a^2 \) structure. Here, we use it to simplify the integral calculation by recognizing that the solution is aligned with the general formula involving natural logarithms as highlighted in pre-calculated identities.

Integration Techniques

Finding the integral of a function can often require different strategies, such as substitution, part by part integration, or applying known identities. In our problem, using a known integral identity streamlines the process significantly. One crucial technique is pattern recognition within the integral, which allows us to map the problem onto a known formula. In our case, recognizing the pattern \( \int \frac{1}{\sqrt{x^2 - a^2}} \, dx \) and knowing the identity \( \ln |x + \sqrt{x^{2}-a^{2}}| + C \) allowed us to directly solve the integral without further transformation.This approach saves time and reduces complexity, avoiding unnecessary steps that might complicate the calculation.

Definite Integrals

While our exercise involved an indefinite integral, understanding definite integrals is equally important in calculus. A definite integral calculates the net area under a curve between two specific points. This differs from an indefinite integral which represents a family of functions.Definite integrals provide concrete values and are expressed as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) define the limits of integration. To solve a definite integral, we generally determine the antiderivative or the indefinite integral first, and then evaluate it at the upper and lower limits \( a \) and \( b \). The result provides the exact area or exact net change over the specified domain. This is a powerful tool for applications such as physics and engineering, where precise measurements are crucial.