Question

For the following exercises, evaluate the integral using the specified method.\(\int \frac{3 x}{x^{3}+2 x^{2}-5 x-6} d x\) using partial fractions

Short answer

\[\int \frac{3 x}{x^{3}+2 x^{2}-5 x-6} d x = \ln|x+1| + 2\ln|x-2| + C\]

Step-by-step solution

Factor the Denominator

The first step is to factor the polynomial in the denominator, \(x^3 + 2x^2 - 5x - 6\). By trial and error, synthetic division or similar techniques, determine the factors. The factored form is \((x + 1)(x - 2)(x + 3)\).

Set Up Partial Fraction Decomposition

Express the integrand using partial fractions: \(\frac{3x}{(x+1)(x-2)(x+3)} = \frac{A}{x+1} + \frac{B}{x-2} + \frac{C}{x+3}\).

Solve for Coefficients

Multiply through by the common denominator \((x+1)(x-2)(x+3)\) to clear the fractions. This results in \(3x = A(x-2)(x+3) + B(x+1)(x+3) + C(x+1)(x-2)\). Expand and collect like terms to generate a system of equations to solve for A, B, and C. Solving gives \(A = 1, B = 2, C = 0\).

Integrate Each Term Separately

Now integrate the expression: \(\int \left( \frac{1}{x+1} + \frac{2}{x-2} \right) dx\). This breaks into two simpler integrals: \(\int \frac{1}{x+1} dx + 2\int \frac{1}{x-2} dx\).

Apply Logarithm Integration

Each integral can be solved using the natural logarithm: \(\int \frac{1}{x+1} dx = \ln|x+1|\) and \(2\int \frac{1}{x-2} dx = 2\ln|x-2|\). Thus, the solution is \(\ln|x+1| + 2\ln|x-2| + C\).

Key concepts

Partial Fraction Decomposition

Partial fraction decomposition is a crucial technique in integral calculus, especially when dealing with rational functions. It allows us to break down a complex fraction into simpler fractions, making integration much easier. Here's a simple breakdown of how it works:

• Identify the polynomial in the denominator that needs decomposition. In our example, this is the cubic polynomial \(x^3 + 2x^2 - 5x - 6\).
• Factor the polynomial fully, if possible. Once factored, you can express the original fraction as a sum of simpler fractions. For instance, after factoring, our polynomial becomes \((x+1)(x-2)(x+3)\).
• Determine the form of the partial fractions. For each factor of the denominator, introduce a constant numerator. This transforms the original fraction into \(\frac{A}{x+1} + \frac{B}{x-2} + \frac{C}{x+3}\).
• Clear the fractions by multiplying through by the common denominator and solve for these constants. This step is crucial to ensure the original fraction's equality with the decomposed form.

By following these steps, we simplify complex integrals and prepare them for straightforward integration techniques.

Integration Techniques

Integration techniques are methods applied to compute integrals, with each technique tailored to handle specific types of integrands. When integrating, choosing the right method is key:

• Substitution: Useful when the integral contains a function and its derivative. It's like the reverse process of the chain rule.
• Integration by Parts: This is ideal for products of functions, relying on the product rule for derivatives.
• Partial Fraction Decomposition: As shown in our exercise, this technique handles rational functions by transforming them into manageable pieces.
• Trigonometric Integrals: Involves integrals of trigonometric functions, often requiring specific identities or transformations.

Each technique has its own set of rules and steps, so understanding their applications and limits is important for efficiently solving integrals. Our exercise used partial fractions because it simplifies the rational function into easier integrals to tackle.

Polynomial Factoring

Polynomial factoring is an essential mathematical skill needed for techniques like partial fraction decomposition. It involves breaking down a polynomial into simpler multiplicative components, called factors.

• Start by looking for the greatest common factor (GCF) in the polynomial terms.
• If the polynomial is quadratic or higher degree, apply methods like synthetic division, trial and error, or the quadratic formula if applicable.
• For our example, the polynomial \(x^3 + 2x^2 - 5x - 6\) can be factored into \((x+1)(x-2)(x+3)\).

Finding these factors is crucial for partial fractions, as they allow us to decompose the fraction correctly. A well-practiced understanding of factoring speeds up the entire process of integral calculus significantly.

Natural Logarithmic Integration

Natural logarithmic integration is an elegant technique used when integrating functions of the form \(\frac{1}{x}\). It's often the result of decomposed fractions achieved after using partial fraction decomposition.

• The integral of \(\frac{1}{x}\) is straightforward, yielding \(\ln|x|\).
• When dealing with terms like \(\frac{1}{x+a}\) or \(\frac{1}{x-a}\), the result is \(\ln|x+a|\) or \(\ln|x-a|\), respectively.
• Our exercise demonstrated this with \(\frac{1}{x+1}\) and \(\frac{2}{x-2}\), leading to \(\ln|x+1|\) and \(2\ln|x-2|\).

Natural logarithmic integration is particularly smooth as it converts integrals into logarithmic expressions. This connection between calculus and logarithms shows how one can elegantly move from a seemingly complex calculation to a natural and simple one.