Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{2}{x^{2}+3 x} d x\)

Short answer

\(\int \frac{2}{x^{2}+3 x} dx = \frac{2}{3} \ln \left|\frac{x}{x+3}\right| + C\).

Step-by-step solution

Factor the Denominator

The denominator of the integrand \(x^2 + 3x\) can be factored as \(x(x + 3)\). This helps set up the partial fraction decomposition.

Set Up Partial Fraction Decomposition

We express \(\frac{2}{x(x+3)}\) as the sum of two partial fractions: \(\frac{2}{x(x+3)} = \frac{A}{x} + \frac{B}{x+3}\).

Solve for Coefficients A and B

Combine the right side into a single fraction: \(\frac{A(x+3) + Bx}{x(x+3)} = \frac{2}{x(x+3)}\). We equate numerators: \(A(x+3) + Bx = 2\).

Determine Values for A and B

Expanding the expression \(A(x+3) + Bx = Ax + 3A + Bx = 2\) leads to \((A + B)x + 3A = 2\). By comparing coefficients, we get \(A + B = 0\) and \(3A = 2\). Solving these, \(A = \frac{2}{3}\) and \(B = -\frac{2}{3}\).

Rewrite the Integral

With the coefficients found, rewrite the integral: \(\int \frac{2}{x(x+3)} dx = \int \left(\frac{2/3}{x} - \frac{2/3}{x+3}\right) dx\).

Integrate Each Term

Integrate the terms separately: \(\int \frac{2/3}{x} dx = \frac{2}{3} \ln |x| + C_1\) and \(\int \frac{-2/3}{x+3} dx = -\frac{2}{3} \ln |x+3| + C_2\).

Combine Results and Simplify

Combine the results with a single constant of integration: \(\frac{2}{3} \ln |x| - \frac{2}{3} \ln |x+3| + C\). Use the logarithmic property \(\ln a - \ln b = \ln \frac{a}{b}\) to simplify: \(\frac{2}{3} \ln \left|\frac{x}{x+3}\right| + C\).

Key concepts

Integral Calculus

Integral calculus is an essential branch of mathematics focused on the concept of integration, which is essentially the opposite of differentiation. It helps calculate areas under curves, solve differential equations, and determine total values in various scenarios. In this exercise, we are tasked with integrating the function \( \frac{2}{x^{2}+3x} \ d x\), which relates to finding the accumulated value described by the given expression.
When dealing with complex functions that form part of a fraction, like in our exercise, direct integration isn't always simple. This is where techniques like partial fraction decomposition come into play to simplify the process. Since integration is about accumulation, it's crucial that we apply methods allowing us to easily add up every small piece that makes up the function's area under the curve.

Factoring Polynomials

Factoring polynomials is a fundamental technique used to simplify expressions, especially when integrating complex functions. In this exercise, we encounter the expression \(x^2 + 3x\) in the denominator, making direct integration difficult. Breaking this down involves recognizing it as a polynomial that can be factored into simpler parts.

To factor \(x^2 + 3x\), you start by identifying common factors; in this case, \(x\). Thus, the expression can be rewritten as \(x(x+3)\). This decomposition is vital because it transforms the denominator into a product of linear terms, setting the stage for the next procedure, partial fraction decomposition. Factoring simplifies complex algebraic expressions and is particularly useful in calculus when dealing with integrals involving polynomials.

Logarithmic Integration

Logarithmic integration is a method used specifically when integrating functions in the form \(\frac{1}{x}\). It's a handy tool, especially after you've broken down a complex fraction into simpler parts using partial fraction decomposition. In our exercise, we apply this technique once we have the decomposition.

In the integral \( \int \frac{2}{x} - \frac{2}{x+3} \ dx \), each term corresponds to a logarithmic function. For example, integrating \( \frac{2}{3x} \) results in the expression \(\frac{2}{3} \ln |x|\), while integrating \(-\frac{2}{3(x+3)}\) gives \(-\frac{2}{3} \ln |x+3|\). Logarithmic functions are particularly nice in integrals because they succinctly represent accumulated growth or decay and can simplify the results using properties of logarithms.

Partial Fractions

The method of partial fractions is a technique used to break down complex rational expressions into simpler, more manageable parts. This approach is particularly useful in integral calculus when dealing with rational functions, where the denominator, like \(x(x+3)\) in our example, is composed of simpler linear factors.

The goal is to express \( \frac{2}{x(x+3)} \) as a sum of fractions using unknown coefficients \(A\) and \(B\). Thus, we write \(\frac{2}{x(x+3)} = \frac{A}{x} + \frac{B}{x+3}\). By solving the equations derived from equating coefficients, we find specific values for \(A\) and \(B\) and simplify the complex fraction into parts that are straightforward for integration.

• Decompose the fraction into simpler fractions.
• Solve for unknown coefficients.
• Use these coefficients to rewrite the integral more simply.

This process simplifies the integration challenge by making each piece of the fraction easy to integrate, resulting in a solution that is straightforward and reduces computation complexity.