Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{17 x-3}{3 x^{2}+x-2} d x\)

Short answer

The integral is \(\frac{7}{3}\ln |3x - 2| + 5\ln |x + 1| + C\).

Step-by-step solution

Factor the Denominator

First, we need to factor the quadratic expression in the denominator, \(3x^2 + x - 2\). To factor this quadratic expression, we find two numbers that multiply to \(-6\) (the product of the coefficient of \(x^2\) and the constant term) and add to \(1\) (the coefficient of \(x\)). The numbers are \(3\) and \(-2\). So, the factorization is \[3x^2 + x - 2 = (3x - 2)(x + 1)\].

Setup Partial Fraction Decomposition

Now, we set up the fraction \(\frac{17x-3}{(3x - 2)(x + 1)}\) as a sum of partial fractions. We write:\[\frac{17x-3}{(3x - 2)(x + 1)} = \frac{A}{3x - 2} + \frac{B}{x + 1}\]where \(A\) and \(B\) are constants to be determined.

Solve for Constants

To find \(A\) and \(B\), multiply both sides by the denominator \((3x - 2)(x + 1)\) to clear the fractions:\[17x - 3 = A(x + 1) + B(3x - 2)\]Expanding the right side, we have:\[17x - 3 = Ax + A + 3Bx - 2B\]Combine like terms:\[(A + 3B)x + (A - 2B)\]Set up the equations from coefficients: 1. \(A + 3B = 17\)2. \(A - 2B = -3\)Solve this system of equations to find \(A\) and \(B\).

Solve the System of Equations

We have the system:\[A + 3B = 17\]\[A - 2B = -3\]Add the two equations to eliminate \(A\):\[2A = 17 - 3B + A - 2B\]\[2A = 14\] \\[A = 7\]Substitute \(A = 7\) back into one of the equations, \(A - 2B = -3\):\[7 - 2B = -3\]\[2B = 10\] \\[B = 5\]

Integrate the Partial Fractions

Now substitute \(A = 7\) and \(B = 5\) into the partial fractions:\[\int \frac{7}{3x - 2} \,dx + \int \frac{5}{x + 1} \,dx\]The antiderivative of these are found using logarithmic integration:\[\frac{7}{3} \ln |3x - 2| + 5\ln |x + 1| + C\]where \(C\) is the constant of integration.

Key concepts

Integration Techniques

Partial Fraction Decomposition is a powerful method used to integrate rational functions. A rational function is a fraction where both the numerator and the denominator are polynomials. With this technique, we break down a complex fraction into simpler fractions that can be easily integrated.
By expressing complex fractions as a sum of simpler fractions, we can apply basic integration formulas to solve the integral.
This technique is especially handy when the denominator is factored into linear or irreducible quadratic factors, making it easier to work with integrals that would otherwise be complicated.

Quadratic Factorization

To perform Partial Fraction Decomposition effectively, it's crucial to factor the denominator of the rational expression. When we come across a quadratic in the denominator, factorization is the first step.
Take the quadratic expression in the exercise: \[3x^2 + x - 2.\]
We look for two numbers that multiply to the product of the leading coefficient and the constant term, which is \{-6\}, and add up to the middle coefficient, which is \{1\}.
By identifying these numbers, we find the correct factors, \{(3x - 2)(x + 1)\}, allowing us to proceed with Partial Fraction Decomposition.

Logarithmic Integration

Once we have decomposed the rational expression into partial fractions, the next step is integration.
The given fractions \(\frac{A}{3x - 2}\) and \(\frac{B}{x + 1}\) are integrated using logarithmic integration rules.
The integral of a fraction of the form \(\frac{A}{ax + b}\) is \(\frac{A}{a}\ln |ax + b| + C\), where \{C\} is the constant of integration.
Applying this to our partial fractions, we simplify the process and arrive at the final answer.
Logarithmic integration is a straightforward technique once the fractions are simplified, turning complex rational functions into log functions.

System of Equations

To find the constants in the Partial Fraction Decomposition, solving a system of equations is necessary.
After setting up the equation based on the partial fraction format, equate coefficients of the same powers of \(x\) from both sides to construct equations.
In our case, solving the system:

• \(A + 3B = 17\)
• \(A - 2B = -3\)

helps us find the values of \(A\) and \(B\) for decomposition.
Basic algebraic techniques, like elimination or substitution, can be applied to derive these constants, making the Partial Fraction Decomposition possible and paving the way for more advanced calculations.