Problem 23 - University Calculus: Early Transcendentals

Question

Evaluate the integrals. $$\int \sec ^{2}(3 x+2) d x$$

Step-by-Step Solution

Answer

\( \frac{1}{3} \tan(3x + 2) + C \) is the antiderivative.

Step 1: Identify the Integral

The integral to be solved is \( \int \sec^2(3x + 2) \, dx \). Notice that the integrand is \( \sec^2(u) \) where \( u = 3x + 2 \), which suggests the antiderivative will involve the tangent function.

Step 2: Make a Substitution

To simplify the integral, we perform a substitution. Let \( u = 3x + 2 \). Then, differentiate \( u \) with respect to \( x \) to find \( du = 3 \, dx \). Solve for \( dx \): \( dx = \frac{1}{3} \, du \).

Step 3: Substitute and Simplify

Replace \( 3x + 2 \) with \( u \) and \( dx \) with \( \frac{1}{3} \, du \) in the integral. The integral becomes \( \int \sec^2(u) \cdot \frac{1}{3} \, du = \frac{1}{3} \int \sec^2(u) \, du \).

Step 4: Integrate Using Known Antiderivative

The integral of \( \sec^2(u) \) is a standard result: \( \int \sec^2(u) \, du = \tan(u) + C \), where \( C \) is the constant of integration. Therefore, \( \frac{1}{3} \int \sec^2(u) \, du = \frac{1}{3} \tan(u) + C \).

Step 5: Substitute Back in Terms of x

Replace \( u \) with the original expression \( 3x + 2 \) to express the integral in terms of \( x \). Thus, the final solution is \( \frac{1}{3} \tan(3x + 2) + C \).

Key Concepts

Trigonometric Integrals

Trigonometric integrals often involve functions like sine, cosine, tangent, and secant. In this exercise, we are dealing specifically with the integral of \( \sec^2(u) \). Recognizing the integrand is crucial as it helps us link it to its antiderivative. The function \( \sec^2(u) \) is one of the basic trigonometric identities that leads to the tangent function. When integrating such trigonometric expressions, it is important to remember these standard results to simplify the process.

Key trigonometric integrals:

• \( \int \sin(u) \, du = -\cos(u) + C \)
• \( \int \cos(u) \, du = \sin(u) + C \)
• \( \int \sec^2(u) \, du = \tan(u) + C \)
• \( \int \csc^2(u) \, du = -\cot(u) + C \)

These relationships are derived from the derivatives of their respective trigonometric functions. Understanding them enables you to quickly identify the antiderivative and integrate efficiently.

Substitution Method

The substitution method is a powerful tool in calculus, especially useful when dealing with integrals that are not straightforward. In this example, the function \( 3x + 2 \) inside \( \sec^2(3x + 2) \) complicates direct integration. To handle this, we perform a substitution to simplify the integral.

Here's the process step-by-step:

• First, identify a part of the integral to substitute. Here, let \( u = 3x + 2 \).
• Next, differentiate \( u \) with respect to \( x \) to find \( du \). This helps find what \( dx \) equals in terms of \( du \), yielding \( du = 3 \, dx \) or \( dx = \frac{1}{3} \, du \).
• Replace all \( x \)-dependent terms in the integral with \( u \)-dependent terms. This swaps the heavy lifting of integration for a simpler problem involving \( u \).

This method is particularly effective for integrals that resemble a function's derivative. Once integrated in terms of \( u \), we replace \( u \) with the original expression in terms of \( x \).

Antiderivative

Finding an antiderivative means discovering a function whose derivative returns the original function. For the function \( \sec^2(u) \), the antiderivative is \( \tan(u) \), because the derivative of \( \tan(u) \) is \( \sec^2(u) \). Knowing these standard antiderivatives by heart makes integration much simpler by allowing us to directly apply these results.

Remember:

• The antiderivative of \( f(u) = \sec^2(u) \) is \( F(u) = \tan(u) + C \).
• Where \( C \) is the constant of integration, representing an infinite number of equivalent functions differing only by a constant.

Once you find the antiderivative, return to the original variable (in this example, \( x \) instead of \( u \)) by substituting back the expression used in the substitution method. This gives us the final solution, ready to be used in further calculations or real-world applications.