Problem 45 - Calculus Single Variable

Question

Calculate each of the indefinite integrals in Exercises \(45-48\). $$ \int \frac{4 x^{2}+x+2}{x^{3}(x+2)} d x $$

Step-by-Step Solution

Answer

Use partial fraction decomposition and integrate each term; combine the results with a constant of integration.

Step 1: Simplifying the Integrand

The given integral is \( \int \frac{4x^2 + x + 2}{x^3(x+2)} \, dx \). First, we simplify the integrand by splitting it into partial fractions. We express it as \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x+2} \).

Step 2: Setting Up the Partial Fraction Equation

Multiply both sides by the denominator \( x^3(x+2) \) to eliminate the fraction. We get the identity:\[4x^2 + x + 2 = A x^2(x+2) + B x(x+2) + C(x+2) + D x^3\].Expand and combine like terms.

Step 3: Solving for Constants

Equate coefficients from both sides of the expanded equation:- \( D x^3 \) must have a coefficient of 0: \( D = 0 \).- For \( x^2 \), we equate \( 4 = A + B \).- For \( x \), we equate \( 1 = 2A + B + C \).- For constants: \( 2 = 2B + C \).Solve the system of equations to find numerical values for \( A, B, C, \) and \( D \).

Step 4: Integrating the Partial Fractions

Substitute the values of \( A, B, C, \) and \( D \) back into our partial fractions setup. Calculate each integral separately:\[\int \frac{A}{x} \, dx, \quad \int \frac{B}{x^2} \, dx, \quad \int \frac{C}{x^3} \, dx, \quad \text{and} \quad \int \frac{D}{x+2} \, dx\].Use known antiderivatives for each term.

Step 5: Writing the Final Solution

Collect all the integrated results and sum them up:\[\int \frac{4x^2 + x + 2}{x^3(x+2)} \, dx = A \ln |x| - \frac{B}{x} - \frac{C}{2x^2} + D \ln |x+2| + C\]Insert the previously calculated constants \( A, B, C, \) and \( D \). Include the constant of integration in the final answer.

Key Concepts

Partial Fractions

Partial fractions is a method used to break down complex rational functions into simpler, easily integrable parts. The trick is to express a complicated fraction as a sum of simpler fractions. In this exercise, we start with the integrand \(\frac{4x^2 + x + 2}{x^3(x+2)}\).
We express this as a sum of fractions with unknown coefficients:

• \(\frac{A}{x}\)
• \(\frac{B}{x^2}\)
• \(\frac{C}{x^3}\)
• \(\frac{D}{x+2}\)

The process of converting the original function into these components is crucial as it allows us to transform an unwieldy function into several basic fractions that are much easier to integrate later.
The real challenge is identifying the correct coefficients \(A\), \(B\), \(C\), and \(D\), which require some algebraic manipulation skills and understanding of how polynomial terms interact.

Antiderivatives

Antiderivatives are a fundamental concept in calculus, involving the reverse process of differentiation. Once the complex fraction is split into partial fractions, the next step is computing the antiderivative of each part. For this exercise:

• \(\int \frac{A}{x} \, dx = A \ln |x| + C\)
• \(\int \frac{B}{x^2} \, dx = -\frac{B}{x} + C\)
• \(\int \frac{C}{x^3} \, dx = -\frac{C}{2x^2} + C\)
• \(\int \frac{D}{x+2} \, dx = D \ln |x+2| + C\)

These simple forms are easier to integrate by recognizing standard integral rules, such as the integral of \(1/x\), and powers of \(x\).
Understanding these rules makes it more straightforward to compute the integral of any similar expressions encountered in calculus.

Simplifying Integrands

Simplifying the integrand is often necessary to make problems more manageable and approachable. In the given exercise, simplification involves expressing the initial complex rational function as partial fractions. This process reduces the cumbersome expression into smaller parts, which are easier to handle.

The strategy of breaking down an integrand into partial fractions allows us to transform difficult integrals into a sum of simpler terms. Each term can often be recognized directly or adapted into standard calculus rules, making the integration process smoother and less error-prone.
Understanding how to simplify an integrand effectively is a valuable skill, as it reduces the potential for mistakes when performing integration.

Equating Coefficients

Equating coefficients is a technique used to find the unknown coefficients in partial fractions. After transforming the original integrand to \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x+2}\), you need to multiply both sides by the common denominator \(x^3(x+2)\) to get rid of the fractions.

This gives an equation where you expand, combine terms, and then compare coefficients of like powers of \(x\). For example, in this exercise:

• Coefficient of \(x^3\) gives \(D = 0\)
• Coefficient of \(x^2\) gives \(A + B = 4\)
• Coefficient of \(x\) gives \(2A + B + C = 1\)
• Constant term gives \(2B + C = 2\)

Setting up and solving these equations is a methodical way to find numerical values for \(A\), \(B\), \(C\), and \(D\), which then allows us to complete the integration effectively. The equating coefficients method aids in systematically working through complex algebraic identities.