Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-54, perform the indicated integrations. \(\int \frac{(6 t-1) \sin \sqrt{3 t^{2}-t-1}}{\sqrt{3 t^{2}-t-1}} d t\)

Step-by-Step Solution

Answer

The integral evaluates to \(-\cos(\sqrt{3t^2 - t - 1}) + C\)."

Step 1: Identify the Integration Technique

The integral given is \( \int \frac{(6t-1) \sin \sqrt{3t^2-t-1}}{\sqrt{3t^2-t-1}} \, dt \). The presence of the square root \( \sqrt{3t^2-t-1} \) and its derivative \( 6t-1 \) in the numerator suggests a substitution method. Specifically, we can use a substitution where the variable inside the square root is set as a new variable.

Step 2: Perform the Substitution

Let \( u = \sqrt{3t^2-t-1} \). Then, squaring both sides gives \( u^2 = 3t^2 - t - 1 \). Taking derivatives yields \( 2u \frac{du}{dt} = 6t - 1 \). Therefore, \( dt = \frac{2u}{6t-1} \, du \). However, notice that the expression \( 6t-1 \) precisely matches the term needed for substitution.

Step 3: Substitute and Simplify the Integral

Substitute \( u = \sqrt{3t^2 - t - 1} \) into the integral. The original integral becomes \( \int \frac{(6t-1) \sin u}{u} \cdot \frac{du}{6t-1} \). The terms \( 6t-1 \) cancel out, simplifying the integral to \( \int \sin u \, du \).

Step 4: Integrate the Simplified Expression

The integral \( \int \sin u \, du \) is a basic integral. Its solution is \(-\cos u + C\), where \( C \) is the constant of integration.

Step 5: Back-Substitute the Original Variable

Replace \( u \) back with the original variable substitution \( u = \sqrt{3t^2 - t - 1} \). Thus, the antiderivative becomes \(-\cos(\sqrt{3t^2 - t - 1}) + C\).

Key Concepts

Substitution Method

The substitution method is a valuable tool in calculus for simplifying complex integrals. It's particularly useful when dealing with integrals involving composite functions. The idea is to substitute part of the integral with a new variable. This turns a daunting integral into a simpler one.

• Identify the part of the integral: Look for a function within another, such as a square root, that also appears in the derivative form. For our example, we have the expression \( \sqrt{3t^2 - t - 1} \) whose derivative, \( 6t-1 \), is conveniently positioned in the numerator.
• Define a new variable: Set \( u = \sqrt{3t^2 - t - 1} \). This change of variables aims to reduce the complexity.
• Rework the integral: Substitute \( u \) into the integral, changing all expressions of \( t \) to expressions of \( u \). This usually simplifies the integral considerably.
• Revert the substitution: After integration, don't forget to convert back to the original variable to get the final antiderivative.

Substitution is powerful because it transforms an intricate integral into a manageable one, bringing solutions within reach.

Indefinite Integral

An indefinite integral is the reverse process of differentiation and represents a family of functions whose derivatives yield the integrand. These integrals are expressed generally with a constant of integration, \( C \), because differentiation does not indicate the original constant that might have existed.

• Symbolizing the process: The notation \( \int f(x) \, dx \) denotes finding the antiderivative of \( f(x) \).
• Including the constant of integration: Since derivatives of constants vanish, recovered functions from integration are added with \( C \).
• Example: The integral \( \int \sin u \, du = -\cos u + C \) demonstrates how a basic pattern leads to another function plus \( C \).

Indefinite integrals can describe curves and areas dynamically since adding \( C \) shifts the curve vertically yet maintains its shape.

Trigonometric Integration

Trigonometric integration frequently appears when dealing with sine, cosine, and other trigonometric functions within integrals. Recognizing and utilizing trigonometric identities can simplify these processes.

• Basic integrals: Familiarity with integrals like \( \int \sin x \, dx = -\cos x + C \) and \( \int \cos x \, dx = \sin x + C \) is essential.
• Using identities: Trigonometric identities, such as \( \sin^2 x + \cos^2 x = 1 \), help replace or reduce expressions to simplify integration.
• Simplifying complex expressions: For complex trigonometric integrals, substitution might involve setting a trigonometric function as the new variable, as shown in our specifc example where integration reduces to \( \int \sin u \, du \).

Mastering trigonometric integration is pivotal as it expands the capability to tackle a wide array of calculus problems, making intricate problems approachable.