Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x-11}{x^{2}+3 x-4} d x\)

Short answer

The integral is \(3 \ln |x+4| - 2 \ln |x-1| + C\).

Step-by-step solution

Factor the Denominator

To perform a partial fraction decomposition, we first need to factor the quadratic expression in the denominator. The expression is \(x^2 + 3x - 4\). This can be factored into \((x + 4)(x - 1)\).

Setup Partial Fraction Decomposition

Once the denominator is factored, write the fraction \(\frac{x-11}{x^2+3x-4} = \frac{x-11}{(x+4)(x-1)}\) as a sum of two simpler fractions: \(\frac{A}{x+4} + \frac{B}{x-1}\).

Clear the Denominator

Multiply through by the original denominator \((x+4)(x-1)\) to obtain the equation \(x - 11 = A(x - 1) + B(x + 4)\).

Expand and Solve for A and B

Expand and rearrange: \(x - 11 = Ax - A + Bx + 4B\), which simplifies to \((A + B)x + (-A + 4B) = x - 11\). By equating coefficients, we get the system of equations: \(A + B = 1\) and \(-A + 4B = -11\).

Solve the System of Equations

Solve the system: \(A + B = 1\) and \(-A + 4B = -11\). From the first equation, \(A = 1 - B\). Substituting into the second equation gives: \(-1 + B + 4B = -11\) which simplifies to \(5B = -10\) leading to \(B = -2\). Substituting \(B = -2\) into \(A = 1 - B\) gives \(A = 3\).

Rewrite the Integral with Partial Fractions

Substitute \(A\) and \(B\) back into the decomposition: \(\frac{3}{x+4} - \frac{2}{x-1}\). The integral becomes \(\int \left( \frac{3}{x+4} - \frac{2}{x-1} \right) dx\).

Integrate Each Term Separately

Integrate the terms separately: \(\int \frac{3}{x+4} dx - \int \frac{2}{x-1} dx\). This gives \(3 \ln |x+4| - 2 \ln |x-1| + C\), where \(C\) is the constant of integration.

Key concepts

Integration Techniques

Integration techniques are essential tools in calculus, allowing us to find the antiderivative or the area under a curve. One powerful technique that often arises with rational functions is partial fraction decomposition. This technique involves breaking down a complex fraction into simpler components that are easier to integrate. It relies on decomposing the original fraction into a sum of fractions with simpler denominators. For instance, when faced with a fraction like \(\frac{x-11}{x^2+3x-4}\), partial fraction decomposition simplifies integration.

### Steps in Partial Fraction Decomposition

• Factor the denominator if possible. In this example, \(x^2+3x-4\) factors into \((x+4)(x-1)\).
• Set up an equation where the original fraction equals a sum of fractions with unknown coefficients. Here, we write \(\frac{A}{x+4} + \frac{B}{x-1}\).
• Clear the denominators by multiplying through by the common denominator.
• Solve for the unknown coefficients by equating coefficients of like terms and solving the resulting linear equations.

With these coefficients, rewrite the original integral as a sum of simpler integrals and proceed with integration. This approach transforms a challenging integral into a series of straightforward ones.

Calculus Problem Solving

Solving calculus problems often involves multiple steps and an understanding of various techniques. Let's break down how to approach a typical integration problem using partial fraction decomposition.

### Approach to Problem Solving

• Identify the type of function: Recognize if the function can be decomposed into simpler terms.
• Factor the denominator: Factoring is crucial in setting up the partial fractions.
• Set up and solve: Write the decomposition, clear denominators, and solve for unknowns by comparing coefficients.
• Integrate term by term: Integrate each partial fraction separately to find the antiderivative.

By deconstructing the process into these logical steps, each integration becomes manageable. Each problem is an opportunity to apply these fundamental techniques effectively, ensuring a thorough understanding of the underlying calculus concepts.

Quadratic Factoring

Quadratic factoring is a mathematical method used to rewrite quadratic expressions. A crucial skill for tackling integration problems involving partial fractions, this technique is applied when the denominator of our fraction is a quadratic expression. Quadratic expressions take the form \(ax^2 + bx + c\) and may factor into two binomial expressions.

### How to Factor Quadratics

• Look for common factors: Begin by checking if there's a common factor in all terms.
• Use factor pairs: Find two numbers that multiply to \(ac\) (the product of the quadratic's leading coefficient and constant term) and add to \(b\).
• Rewrite and factor: Use the numbers found to split the middle term, then factor by grouping.

In the exercise, \(x^2 + 3x - 4\) factors into \((x+4)(x-1)\). Recognizing these factors allows us to set up the partial fractions necessary for integration. A solid understanding of quadratic factoring eases the process of tackling more complex calculus challenges.