Problem 16 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-28\). $$ \int \frac{d x}{x^{2} \sqrt{x^{2}+1}} $$

Step-by-Step Solution

Answer

The integral evaluates to \(-\frac{1}{x} + C\).

Step 1: Identify the substitution

To solve the integral \( \int \frac{d x}{x^{2} \sqrt{x^{2}+1}} \), we should recognize that the presence of \( \sqrt{x^2+1} \) suggests a trigonometric substitution. Specifically, we use \( x = \tan(\theta) \), which implies that \( dx = \sec^2(\theta) \, d\theta \) and \( \sqrt{x^2 + 1} = \sec(\theta) \).

Step 2: Apply the substitution

Substituting \( x = \tan(\theta) \), the differential becomes \( dx = \sec^2(\theta) \, d\theta \), and the integral is rewritten in terms of \( \theta \): \[ \int \frac{\sec^2(\theta) \, d\theta}{\tan^2(\theta) \sec(\theta)}. \] This simplifies to: \[ \int \frac{\sec(\theta) \, d\theta}{\sin^2(\theta)} = \int \csc(\theta) \sec(\theta) \, d\theta. \]

Step 3: Simplify and evaluate the integral

The rewritten integral \( \int \csc(\theta) \sec(\theta) \, d\theta \) suggests using trigonometric identities. Rewrite \( \csc(\theta) = \frac{1}{\sin(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \). The integral becomes: \[ \int \frac{1}{\sin(\theta) \cos(\theta)} \, d\theta = \int \frac{2}{\sin(2\theta)} \, d\theta. \] Using the substitution \( u = \tan(\theta) \) for integration can simplify the process.

Step 4: Solve the integral

Let \( u = \tan(\theta) \), then \( du = \sec^2(\theta) \, d\theta \). The integral becomes: \[ \int \frac{du}{u^2} = -\frac{1}{u} + C. \]

Step 5: Back-substitute to original variable

Returning to the variable \( x \), since \( u = \tan(\theta) = x \), the expression becomes: \(-\frac{1}{x} + C\). Thus, the evaluated integral is: \[ \int \frac{d x}{x^{2} \sqrt{x^2+1}} = -\frac{1}{x} + C. \]

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots, especially those including expressions like \( \sqrt{x^2 + a^2} \), \( \sqrt{x^2 - a^2} \), or \( \sqrt{a^2 - x^2} \). In this exercise, we encounter \( \sqrt{x^2 + 1} \), which is fitting for a trigonometric substitution involving tangent and secant. The idea is to replace the variable with a trigonometric function, specifically using \( x = \tan(\theta) \). This choice transforms \( \sqrt{x^2+1} \) into \( \sec(\theta) \), greatly simplifying the integral.

• The substitution exploits the Pythagorean identity: \( 1 + \tan^2(\theta) = \sec^2(\theta) \).
• This method converts the integral into a form where trigonometric identities and simple integration rules can be applied.
• It effectively reduces a more complex algebraic expression to a simpler trigonometric integral.

Recall that after calculating the integral in terms of \( \theta \), we must return to the original variable \( x \) through back-substitution.

Definite Integrals

Though this example evaluates an indefinite integral, understanding definite integrals is crucial in calculus. Definite integrals deal with evaluating the integral of a function over a specific interval \([a, b]\). This process has geometric implications, as it computes the exact area under the curve described by the function between the given bounds.

• The result of a definite integral is a number, unlike an indefinite integral whose result includes a constant of integration \( C \).
• When calculating, one typically finds the indefinite integral first, then applies the limits \( a \) and \( b \) to compute the specific value.

Here, definite integrals were not directly involved, but understanding them helps in situations where bounds are specified, which often require evaluating such integrals.

Trigonometric Identities

Trigonometric identities simplify calculations in calculus, especially when dealing with integrals involving trigonometric functions. In this problem, identities like \( \sec^2(\theta) = 1 + \tan^2(\theta) \) and \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \) play pivotal roles.

• These identities allow transformations of the integrand into forms that are easier to integrate.
• Simplifying \( \frac{1}{\sin(\theta)\cos(\theta)} \) to \( \frac{2}{\sin(2\theta)} \) using identities reduces the complexity of the integral.

Such identities are invaluable when integrating functions that don't readily simplify using algebra alone, exemplified in the conversion from algebraic terms to trigonometric known integrals.

Calculus Problem Solving

Solving calculus problems often involves recognizing patterns and selecting the appropriate mathematical tool or technique. This exercise showcases the step-by-step problem-solving approach using trigonometric substitution.

• Identifying the presence of a radical expression hints at substitution methods.
• Transforming the problem with substitution transitions it into an easier scenario to handle.
• The ability to switch back to the original variable, after solving in terms of a new one, is crucial for returning to the context of the original problem.

Effective calculus problem solving requires understanding when and how to apply various techniques, like trigonometric substitution, while ensuring a connection is maintained to the original variable throughout the solution process.