Problem 29 - Calculus Single Variable

Question

Use the method of partial fractions to decompose the integrand. Then evaluate the given integral. \(\int \frac{3 x^{3}-5 x^{2}+10 x-19}{\left(x^{2}+4\right)\left(x^{2}+3\right)} d x\)

Step-by-Step Solution

Answer

The integral is \(\frac{1}{2} \ln|x^2 + 4| + \tan^{-1}(\frac{x}{2}) + \ln|x^2 + 3| - 2 \sqrt{3} \tan^{-1}(\frac{x}{\sqrt{3}}) + C\).

Step 1: Analyze the Denominator

The denominator of the integrand is \((x^2 + 4)(x^2 + 3)\). This is a product of two distinct irreducible quadratic factors.

Step 2: Set Up Partial Fractions

Express the integrand as a sum of partial fractions: \[ \frac{3x^{3}-5x^{2}+10x-19}{(x^{2}+4)(x^{2}+3)} = \frac{Ax + B}{x^{2}+4} + \frac{Cx + D}{x^{2}+3}. \]

Step 3: Solve for Coefficients

To find the coefficients \(A\), \(B\), \(C\), and \(D\), equate: \[ 3x^3 - 5x^2 + 10x - 19 = (Ax + B)(x^2 + 3) + (Cx + D)(x^2 + 4). \] Expand and collect like terms to get a system of equations by comparing coefficients of like powers of \(x\).

Step 4: Expand and Collect Terms

Expand: \[(Ax + B)(x^2 + 3) = Ax^3 + 3Ax + Bx^2 + 3B,\]\[(Cx + D)(x^2 + 4) = Cx^3 + 4Cx + Dx^2 + 4D.\]Combine to get:\[(A + C)x^3 + (B + D)x^2 + (3A + 4C)x + (3B + 4D).\]

Step 5: Set Up the Equations

By comparing coefficients from the expanded expression \((A + C)x^3 + (B + D)x^2 + (3A + 4C)x + (3B + 4D)\) with \(3x^3 - 5x^2 + 10x - 19\), write the system:1. \(A + C = 3\)2. \(B + D = -5\)3. \(3A + 4C = 10\)4. \(3B + 4D = -19\)

Step 6: Solve the System of Equations

Solve the system: From equation 1: solve for \(C = 3 - A\). Substitute into equation 3: \[3A + 4(3 - A) = 10\] Solve for \(A = 1\), giving \(C = 2\).From equation 2: \(B + D = -5\). Substitute into equation 4: \[3B + 4(-5 - B) = -19\] Solve for \(B = 1\), giving \(D = -6\).

Step 7: Rewrite the Integral

Substitute the found coefficients back into the partial fractions: \[ \int \left( \frac{x + 1}{x^2 + 4} + \frac{2x - 6}{x^2 + 3} \right) \, dx. \]

Step 8: Integrate the Partial Fractions

Each integral can be solved separately using substitution and known integral formulas:1. \( \int \frac{x + 1}{x^2 + 4} \, dx = \frac{1}{2} \ln|x^2 + 4| + \tan^{-1}\left(\frac{x}{2}\right) + C_1\)2. \( \int \frac{2x}{x^2 + 3} \, dx = \ln|x^2 + 3| + C_2\)3. \( \int \frac{-6}{x^2 + 3} \, dx = -2 \sqrt{3} \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + C_3\)

Step 9: Combine Results and Simplify

Combine the results and add an arbitrary constant \(C\): \[ \frac{1}{2} \ln|x^2 + 4| + \tan^{-1}\left(\frac{x}{2}\right) + \ln|x^2 + 3| - 2 \sqrt{3} \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + C. \]

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematical analysis that focuses on finding the integral, or the accumulation of functions. Integrals can be thought of as the area under a curve or the anti-derivative of a function. In this problem, the aim was to find the integral of a rational function using partial fractions. Understanding integral calculus is vital because it allows us to solve a wide variety of problems, from computing areas and volumes to solving differential equations.

When encountering problems like these, it's important to identify the type of function you are working with. Rational functions, which are the ratio of two polynomials like in this exercise, often require special techniques for integration.

Rational Function Integration

Rational function integration deals with integrating functions expressed as the ratio of two polynomials. Many rational functions can be tricky to integrate directly. In our exercise, the integrand was a rational function \[ \frac{3x^{3}-5x^{2}+10x-19}{(x^{2}+4)(x^{2}+3)}. \]

Using the method of partial fractions, complex rational expressions are decomposed into simpler "partial" fractions that are easier to integrate. The integrand is expressed as a sum of simpler fractions, each of which individually can be integrated using known formulas or techniques. These techniques break the problem into manageable parts, handling complex numerators and denominators systematically.

Breaking down a rational function using partial fractions is crucial in integral calculus as it simplifies the process of finding the antiderivative.

System of Equations

A system of equations is a collection of equations that share common variables. Solving systems of equations provides a solution or set of solutions that satisfy all equations simultaneously. Within this given exercise, we set up a system of four equations to solve for unknown coefficients in the partial fraction decomposition.

These steps involve matching the coefficients from the product of partial fractions with the original rational expression. For example, comparing the expanded polynomial with the original, we obtain a system

• \( A + C = 3 \)
• \( B + D = -5 \)
• \( 3A + 4C = 10 \)
• \( 3B + 4D = -19 \)

Solving systems requires strategies such as substitution or elimination, enabling us to find the unknowns needed for integration. These solutions are essential to write the correct partial fraction decomposition.

Substitution Method

The substitution method is a crucial step in integrating partial fractions. It involves replacing variables in order to simplify the integral and make it more manageable to evaluate. For example, in the final integration phase of the exercise, after having rewritten the integral with our coefficients, we solve them individually by recognizing patterns such as:

• \( \int \frac{x + 1}{x^2 + 4} \, dx \)
• \( \int \frac{2x}{x^2 + 3} \, dx \)
• \( \int \frac{-6}{x^2 + 3} \, dx \)

Using substitution, we could make each part dependent on a simpler function or recognize them as standard forms of integrals.

For instance, the substitution for \( x^2 + a \) type integrals often involves recognizing the derivative in the numerator, making the integral straightforward. Substitution simplifies complex expressions, streamlining the problem-solving process and enhancing understanding of integral calculus.