Problem 33 - Applied Calculus

Question

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{x^{3}+x^{2}}{3 x^{4}+4 x^{3}} d x $$

Step-by-Step Solution

Answer

\( \frac{1}{12} \ln|3x^4 + 4x^3| + C \)

Step 1: Identify the Denominator as a Potential Substitution

Look at the integral \( \int \frac{x^{3}+x^{2}}{3 x^{4}+4 x^{3}} \, dx \). Notice that the denominator \( 3x^4 + 4x^3 \) looks like a good candidate for substitution because its derivative is similar to the numerator.

Step 2: Set Up the Substitution

Let \( u = 3x^4 + 4x^3 \). Calculate the derivative: \( \frac{du}{dx} = 12x^3 + 12x^2 \). This means \( du = (12x^3 + 12x^2) \, dx \).

Step 3: Modify the Integral with Substitution

Rearrange the expression for \( du \) to match the integral's numerator: \( du = 12x^3 + 12x^2 \, dx \). We need to express \( x^3 + x^2 \, dx \), which can be rewritten as \( \frac{1}{12} du \). The integral becomes \( \int \frac{1}{12} \cdot \frac{1}{u} \, du \).

Step 4: Solve the Simplified Integral

Evaluate the integral \( \int \frac{1}{12} \cdot \frac{1}{u} \, du = \frac{1}{12} \int \frac{1}{u} \, du \). The antiderivative of \( \frac{1}{u} \) is \( \ln|u| \). Thus, the solution is \( \frac{1}{12} \ln|u| + C \), where \( C \) is the constant of integration.

Step 5: Substitute Back to Original Variables

Replace \( u \) with the expression in terms of \( x \): \( u = 3x^4 + 4x^3 \). The final answer is \( \frac{1}{12} \ln|3x^4 + 4x^3| + C \).

Key Concepts

Substitution Method

The substitution method is a powerful tool in calculus to simplify integrals. When faced with a complex integral, like \( \int \frac{x^{3}+x^{2}}{3 x^{4}+4 x^{3}} \, dx \), we can try to transform it into something more manageable. The key is identifying a part of the integral, often the denominator, that can be substituted with a new variable, \( u \).

• Choose a Substitution: Look for a function in the integral whose derivative also appears. Here, the denominator \( 3x^4 + 4x^3 \) is chosen because its derivative, \( 12x^3 + 12x^2 \), resembles the numerator.
• Change of Variables: After choosing \( u = 3x^4 + 4x^3 \), differentiate to express \( du \) in terms of \( dx \). In this case, \( du = (12x^3 + 12x^2) \, dx \).
• Reformulate the Integral: Replace all instances of the original variable \( x \) with \( u \). This transforms the integral into \( \int \frac{1}{12} \cdot \frac{1}{u} \, du \).

This method helps break down the integral into simpler parts and demonstrates the beauty of variable transformation in calculus.

Calculus Integration

Integration in calculus is the process of finding the antiderivative or the integral of a function. It is the reverse operation of differentiation. With problems involving integration, the goal is to determine the area under a curve or find a function whose derivative results in the original integrand.

• Indefinite Integrals: These are integrals without specified limits. They represent a family of functions plus an arbitrary constant \( C \). For example, \( \int \frac{1}{u} \, du \) is \( \ln|u| + C \).
• Integration Techniques: There are several techniques to solve integrals, such as substitution, integration by parts, and partial fraction decomposition. The substitution method is particularly helpful with integrals involving composite functions.
• Symbolism and Notation: The integral sign \( \int \) represents integration, and the small \( dx \) indicates the variable of integration. Indefinite integrals don’t have bounds, unlike definite integrals that calculate a numerical value over an interval.

Understanding these basics helps in solving integrals efficiently and recognizing which method fits the problem at hand.

Derivative in Integration

In integration, understanding derivatives is crucial as it allows us to find antiderivatives or integrals. The derivative rules are used to reverse-engineer functions to uncover their integrals. This relationship is often dealt with in the substitution method.

• Connection with Differentiation: The fundamental theorem of calculus establishes the link between differentiation and integration. If you differentiate the result of an integral, you should retrieve the original function within the domain.
• Using Derivatives for Substitution: Recognizing the derivative of a function within the integrand can facilitate the substitution, making it easier to solve the integral. For instance, knowing that \( d(3x^4 + 4x^3)/dx = 12x^3 + 12x^2 \) helps in forming the \( du \) needed in substitution.
• Checking for Errors: Once you've integrated, differentiate your result to check your work. If your derivative matches the original integrand, you've likely integrated correctly. This is a helpful tool for verifying the accuracy of your solution.

By understanding how derivatives factor into integration, students can skillfully maneuver through complex calculus problems with confidence.