Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated integrations. $$ \int \frac{3 x^{2}+2 x}{x+1} d x $$

Step-by-Step Solution

Answer

\(\frac{3}{2}x^2 - x - \ln|x+1| + C\)

Step 1: Simplify the Integrand

The problem involves finding the integral of \( \int \frac{3x^2 + 2x}{x+1} \, dx \). To simplify, use polynomial long division to divide \(3x^2 + 2x\) by \(x + 1\). Performing this division gives a quotient of \(3x - 1\) and a remainder of \(-1\). The integral can now be written as \(\int (3x - 1) + \frac{-1}{x+1} \, dx\).

Step 2: Integrate the Quotient

We can separate the integrals: \(\int (3x - 1) \, dx\) and \(\int \frac{-1}{x+1} \, dx\). First, integrate the polynomial: \(\int (3x - 1) \, dx = \int 3x \, dx - \int 1 \, dx\). The integral of \(3x\) is \(\frac{3}{2}x^2\), and the integral of \(-1\) is \(-x\). Thus, this part of the integral is \(\frac{3}{2}x^2 - x\).

Step 3: Integrate the Remainder

Next, integrate the fraction: \(\int \frac{-1}{x+1} \, dx\). This is a simple natural logarithm integral, resulting in \(-\ln|x+1|\).

Step 4: Combine the Results

Combine the results from Steps 2 and 3 to give the full integrated expression: \(\frac{3}{2}x^2 - x - \ln|x+1| + C\), where \(C\) is the constant of integration.

Key Concepts

Polynomial Long Division

Polynomial long division is a process similar to long division with numbers, but instead of digits, we use polynomial terms. In this exercise, we have the polynomial expression \(3x^2 + 2x\) being divided by \(x + 1\). This technique helps us to simplify complex rational expressions, especially before tackling integration problems.
To perform polynomial long division:

• Divide the leading term of the numerator \(3x^2\) by the leading term of the denominator \(x\), giving you \(3x\).
• Multiply \(3x\) by \(x + 1\), resulting in \(3x^2 + 3x\).
• Subtract that product from the original numerator \(3x^2 + 2x\), resulting in \(-x\).
• Now, divide \(-x\) by \(x\), giving you \(-1\).
• Multiply \(-1\) by \(x + 1\), resulting in \(-x - 1\).
• Subtract that from \(-x\), resulting in a remainder of \(-1\).

The result is a quotient of \(3x - 1\) and a remainder of \(-1\), simplifying our integrand for further steps.

Natural Logarithm Integration

In integration, the natural logarithm is used when integrating expressions like \(\int \frac{1}{x} \, dx\), which results in \(\ln|x|\). Similarly, for a shifted variable \(\int \frac{1}{x+a} \, dx\), the result is \(\ln|x+a|\).
In our exercise, we're focusing on \(\int \frac{-1}{x+1} \, dx\). According to the natural logarithm integration rule, this simplifies to \(-\ln|x+1|\).

Key Points on Natural Logarithm Integration:

• The integral of \(\frac{1}{u}\), where \(u\) is a linear function, results in \(\ln|u|\).
• Make sure to include the constant of integration \(C\), since we're dealing with indefinite integrals.
• The negative sign before \(\ln|x+1|\) is crucial due to the original integral having \(-1\) as its numerator.

Indefinite Integrals

Indefinite integrals are used to find the general form of a function, as opposed to definite integrals which find the area under a curve over a specific interval. In this problem, we denote indefinite integrals using the integral sign \(\int\) without upper and lower limits, producing a function plus an arbitrary constant \(C\).

Concept of Indefinite Integrals:

• Indefinite integrals provide a family of functions rather than a single function.
• Always include a constant \(C\) when determining an indefinite integral, since integration can yield functions differing by a constant.
• Use basic rules, such as \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), when finding the integral of a polynomial.

In our exercise, the solution involves breaking the integral into manageable parts to handle easily using basic integration rules.

Mathematical Problem Solving

Mathematical problem solving in the context of integration involves several key strategies to simplify and work through the problem efficiently. In this integration problem, each step was crucial for arriving at the correct solution.

Approaches for Problem Solving:

• Decomposition: Break down complex expressions into simpler components that are easier to handle individually.
• Simplification: Use techniques like polynomial long division to reduce the complexity of integrands.
• Application of Rules: Recognize when to apply specific integration rules, such as those for natural logarithms.
• Combining Results: After integrating parts of an expression separately, remember to combine these results logically, ensuring all parts contribute to the final answer.

Using these strategies effectively can transform a seemingly tough integration problem into clear, achievable steps, leading to a successful solution.