Problem 35 - Thomas Calculus

Question

In Exercises \(29-36\) , use an appropriate substitution and then a trigonometric substitution to evaluate the integrals. $$ \int \frac{x d x}{\sqrt{x^{2}-1}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Substitution

This integral is suitable for a trigonometric substitution because of the square root of the form \( \sqrt{x^2 - 1} \). Recognizing this as a derivative similar to \( \sec^2\theta - 1 = \tan^2\theta \), we make the substitution \( x = \sec\theta \), so \( dx = \sec\theta \tan\theta \; d\theta \).

Step 2: Substitute and Simplify

Substituting \( x = \sec\theta \) into the integral, we have:\[\int \frac{\sec\theta \tan\theta \; d\theta}{\sqrt{\sec^2\theta - 1}}.\]Since \( \sqrt{\sec^2\theta - 1} = \tan\theta \), the integral simplifies to:\[\int \frac{\sec\theta \tan\theta \; d\theta}{\tan\theta} = \int \sec\theta \; d\theta.\]

Step 3: Integrate

The integral \( \int \sec\theta \; d\theta \) is a standard trigonometric integral. Using the formula \( \int \sec\theta \; d\theta = \ln |\sec\theta + \tan\theta| + C \), we find:\[\int \sec\theta \; d\theta = \ln |\sec\theta + \tan\theta| + C.\]

Step 4: Back-Substitute the Trigonometric Identities

Given \( x = \sec\theta \), it follows that \( \sec\theta = x \) and \( \tan\theta = \sqrt{x^2 - 1} \). Substituting back gives:\[\ln |x + \sqrt{x^2 - 1}| + C.\]Thus, the integral becomes:\[\ln |x + \sqrt{x^2 - 1}| + C.\]

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique employed in integral calculus to simplify integrals involving square roots of quadratic expressions. It's particularly useful when the integrand contains terms like \( \sqrt{a^2 - x^2} \), \( \sqrt{x^2 - a^2} \), or \( \sqrt{x^2 + a^2} \).
This technique works by substituting the variable \( x \) with a trigonometric function. For example, when dealing with \( \sqrt{x^2 - 1} \), as in our original exercise, we use the identity \( 1 + \tan^2\theta = \sec^2\theta \). This substitution simplifies the square root, making the integral more manageable.

• Identify the square root form.
• Select the appropriate trigonometric substitution, such as \( x = \sec\theta \) for \( \sqrt{x^2 - 1} \).
• Convert \( dx \) accordingly and substitute into the integral.

This method simplifies complex expressions and often leads directly to a standard trigonometric integral, which is easier to solve.

Integration Techniques

In calculus, a variety of integration techniques solve different types of integrals. Trigonometric substitution is just one of them. Depending on the problem, you might choose among different methods like substitution, integration by parts, or partial fractions.
In our case, once you use trigonometric substitution, the integral \( \int \sec\theta d\theta \) emerges, which is a standard integration formula. Knowing such formulas is crucial as they allow you to proceed quickly to the solution.

• Integration by substitution involves replacing a part of the integrand to simplify the integral.
• Integration by parts is useful when the integrand is a product of functions, given by \( \int u \, dv = uv - \int v \, du \).
• Partial fraction decomposition is applied to rational functions to break them into simpler fractions.

Always start by analyzing the structure of the integral. This helps determine the best technique to apply and solve efficiently.

Definite Integrals

Definite integrals evaluate the area under a curve between two specific limits. Although the original exercise didn't have specific limits, understanding definite integrals helps grasp the idea of total accumulated quantity.
When working with definite integrals and trigonometric substitution, don’t forget to adjust the limits of integration. If \( x \) changes to \( \theta \) in your substitution, your original limits must reflect this shift, too.

• Definite integrals have both a lower limit \( a \) and an upper limit \( b \), denoted as \( \int_a^b f(x) \, dx \).
• Evaluate the antiderivative at these limits and subtract: \( F(b) - F(a) \).
• Adjust limits when substituting to maintain the correct interval of integration.

Taking the time to carefully substitute and adjust limits ensures precise computation of definite integrals.