Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{6 x^{2}+22 x-23}{(2 x-1)\left(x^{2}+x-6\right)} d x\)

Short answer

Perform partial fraction decomposition and integrate each fraction separately after solving for constants.

Step-by-step solution

Factor the Denominator

First, we need to factor the quadratic expression in the denominator. The given denominator is \((2x-1)(x^2+x-6)\). Let's factor the quadratic \(x^2+x-6\). By using the factoring method, we identify it as \((x+3)(x-2)\). So, the complete factorization of the denominator is \((2x-1)(x+3)(x-2)\).

Set Up the Partial Fractions

With the denominator factored, we set up the partial fraction decomposition as:\[\frac{6x^2 + 22x - 23}{(2x-1)(x+3)(x-2)} = \frac{A}{2x-1} + \frac{B}{x+3} + \frac{C}{x-2}\]where \(A\), \(B\), and \(C\) are constants that need to be determined.

Eliminate the Denominator and Expand

Multiply both sides of the equation by the denominator \((2x-1)(x+3)(x-2)\) to eliminate the fractions:\[6x^2 + 22x - 23 = A(x+3)(x-2) + B(2x-1)(x-2) + C(2x-1)(x+3)\]Next, expand each of the terms on the right-hand side.

Equate and Solve for A, B, C

After expanding, equate the coefficients from both sides of the equation for the same power of \(x\) to create a system of equations:- \(2A + 2B + 2C = 6\) (coefficient of \(x^2\))- \(-A - 3B + 3C = 22\) (coefficient of \(x\))- \(-6A + 2B - 3C = -23\) (constant term)Solve these equations to find the values of \(A\), \(B\), and \(C\).

Integrate Term by Term

Using the values of \(A\), \(B\), and \(C\) found in Step 4, insert them back into the partial fractions:\[\int \left(\frac{A}{2x-1} + \frac{B}{x+3} + \frac{C}{x-2}\right) dx\]Integrate each term separately.- Integrate \(\frac{A}{2x-1}\): Use substitution \(u = 2x-1\).- Integrate \(\frac{B}{x+3}\): Use substitution \(u = x+3\).- Integrate \(\frac{C}{x-2}\): Use substitution \(u = x-2\).

Key concepts

Integration

Integration is a fundamental concept in calculus. It involves finding the integral, or the reverse operation of differentiation. The integral can represent various physical concepts like area under a curve or accumulated change. In indefinite integration, the goal is to find a family of functions that could have been differentiated to give the original function.

For the given exercise, after performing partial fraction decomposition, each fraction is integrated separately. Integrating each term involves using basic techniques, such as substitution, which is a common method to simplify integrations.

Using substitution, you replace a part of the integral with a single variable to simplify the process. Then, you integrate with respect to that variable and substitute back to the original variable. This helps in solving more complex integrals easily.

Algebraic Fractions

Algebraic fractions are expressions that contain polynomials within a numerator and a denominator. Simplifying these fractions, especially involving multiple terms, can often be a trickier task.

In partial fraction decomposition, algebraic fractions are expressed as a sum of simpler fractions. This process helps to integrate complex expressions by breaking them into manageable parts. Identifying the degree of the numerator and denominator is crucial, as it ensures the proper setup for partial fraction decomposition.

• If the degree of the numerator is less than the denominator, direct decomposition is done.
• If the degree of the numerator is equal to or greater than that of the denominator, polynomial long division is first performed.

By expressing an algebraic fraction correctly, integration can be carried out with ease.

Factoring Quadratics

Factoring quadratics involves rewriting the quadratic equation, typically in the form of \(ax^2 + bx + c\), as a product of its factors. It is an essential step in simplifying algebraic fractions and especially useful in partial fraction decomposition.

In the exercise, the quadratic \(x^2 + x - 6\) is factored into \(x+3\) and \(x-2\). To factor a quadratic, you find two numbers that multiply to give the constant term (c) and add up to give the linear coefficient (b).

• Step 1: Set the quadratic in a standard form, if not already.
• Step 2: Identify two numbers based on the product-sum criteria.
• Step 3: Rewrite the middle term using these identified numbers.
• Step 4: Factor by grouping terms if necessary.

Accurate factoring allows for the correct application of the partial fractions method, aiding in solving the integral efficiently.