Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int \cos (3 x+2) d x\)

Step-by-Step Solution

Answer

Substitute \( u = 3x + 2 \), integrate to get \( \int \cos(u) \, du = \sin(u) + C \), and replace \( u \) to get \( \sin(3x+2) + C \).

Step 1: Choose a substitution

Identify the inner function within the composite function \( \cos(3x+2) \).Let \( u = 3x + 2 \).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

Substitute \( u = 3x + 2 \), integrate to get \( \int \cos(u) \, du = \sin(u) + C \), and replace \( u \) to get \( \sin(3x+2) + C \).

Key Concepts

Integration by Substitution

Integration by substitution is a valuable technique in calculus often used to evaluate integrals involving composite functions. The primary goal is to simplify the integral into a more manageable form. You can think of it as a reverse chain rule, where you undo the complication caused by the composition of functions. **How It Works** - **Choose an Inner Function:** Start by identifying a part of the integrand that is part of a composite function. This should be the part that, when differentiated, appears elsewhere in the integrand. For the problem, we choose the substitution {} Let \( u = 3x + 2 \). - **Differentiate the Substitution:** Once you have your substitution, compute its derivative. Here, \( \frac{du}{dx} = 3 \) implies that \( du = 3 \, dx \) or \( dx = \frac{1}{3}du \). - **Rewrite the Integral:** Substitute \( u \) and \( dx \) into the integral. This changes the variable of integration to \( u \), simplifying the expression. The integral \( \int \cos(3x+2) \, dx \) becomes \( \int \cos(u) \cdot \frac{1}{3} \, du \), which simplifies to \( \frac{1}{3} \int \cos(u) \cdot du \). By simplifying the problem using substitution, you can often transform complex integrals into straightforward basic forms.

Composite Function

Composite functions are functions created when one function is applied to the result of another function. They can often complicate the derivation and integration processes. In calculus, this is seen frequently in the context of integration which involves the chain rule in reverse. **Understanding Composite Functions** - **Notation and Structure:** A composite function, denoted as \( f(g(x)) \), applies the function \( f \) to the result of \( g(x) \). In our exercise, \( \cos(3x+2) \) is the composition of the cosine function and the linear function \( 3x+2 \). - **Behavior Under Differentiation:** Differentiating composite functions require the chain rule, where you take the derivative of the outer function evaluated at the inner function and multiply it by the derivative of the inner function. Substitution is used in the opposite process, where identifying u-substitutions simplifies integration. Understanding the structure and behavior of composite functions makes tackling integrals involving such expressions manageable. Recognizing the inner and outer components is key to using substitution effectively.

Trigonometric Integration

Trigonometric integration involves integrating expressions that include trigonometric functions. Such functions include sine, cosine, tangent, and their reciprocals. The goal in such problems is usually to recognize identities, simplify expressions, or use substitution to make the integral solvable. **Integration Involving Cosine** - **Cosine Function Properties:** The integral of cosine, \( \int \cos(x) \, dx \), is straightforward and results in \( \sin(x) + C \), where \( C \) is the constant of integration. - **Using Identities:** In some trigonometric integration problems, identities can be used to transform the integral into a more familiar form. Although it is not needed in this exercise, knowing these identities can be immensely helpful. In our specific exercise, once the substitution reduces the integral to \( \int \cos(u) \, du \), the integration process involves applying the basic integral formula. The result then refers back to the variable \( x \) by substituting \( u = 3x+2 \) back. The solution verifies that using these techniques transforms initially complex trigonometric integrals into simpler forms.