Question

Partial Fractions with Nonrepeated Linear Factors Evaluate \(\int \frac{3 x+2}{x^{3}-x^{2}-2 x} d x\)

Short answer

The integral evaluates to \(\ln\left(\frac{|x-2|^4}{|x||x+1|^3}\right) + C\).

Step-by-step solution

Factor the Denominator

The given fraction is \[ \frac{3x+2}{x^3 - x^2 - 2x} \]First, we need to factor the denominator. Begin by factoring out the common factor \(x\):\[ x^3 - x^2 - 2x = x(x^2 - x - 2) \]Next, factor the quadratic \(x^2 - x - 2\):\[ x^2 - x - 2 = (x - 2)(x + 1) \]Thus, the complete factorization of the denominator is:\[ x(x - 2)(x + 1) \]

Set Up Partial Fraction Decomposition

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

Clear the Denominator

Multiply through by the denominator \(x(x - 2)(x + 1)\) to clear the fractions:\[3x + 2 = A(x - 2)(x + 1) + Bx(x + 1) + Cx(x - 2)\]Simplify and expand the right-hand side.

Expand and Match Coefficients

Expand each term on the right-hand side:- \(A(x - 2)(x + 1) = A(x^2 - x - 2)\)- \(Bx(x + 1) = B(x^2 + x)\)- \(Cx(x - 2) = C(x^2 - 2x)\)Combine like terms:\[ (A + B + C)x^2 + (-A + B - 2C)x - 2A = 3x + 2\]

Solve for Coefficients A, B, and C

Match coefficients from both sides of the equation:- For \(x^2\): \(A + B + C = 0\)- For \(x\): \(-A + B - 2C = 3\)- Constant term: \(-2A = 2\)Solving these equations yields:- \(A = -1\)- Substituting \(A = -1\) in \(-2A = 2\), confirms \(A = -1\).- Substitute \(A = -1\) in \(A + B + C = 0\) gives \(B + C = 1\).- Substitute \(A = -1\) in \(-A + B - 2C = 3\) gives \(1 + B - 2C = 3\), which simplifies to \(B - 2C = 2\).From \(B + C = 1\) and \(B - 2C = 2\), solve to get \(B = 4\) and \(C = -3\).

Write the Integral as a Sum of Partial Fractions

Substitute values for \(A\), \(B\), and \(C\) back into the partial fraction decomposition:\[\int \left(\frac{-1}{x} + \frac{4}{x-2} - \frac{3}{x+1}\right) dx\]

Integrate Each Term Separately

Integrate term by term:- \(\int \frac{-1}{x} \, dx = -\ln|x| + C_1\)- \(\int \frac{4}{x-2} \, dx = 4\ln|x-2| + C_2\)- \(\int \frac{-3}{x+1} \, dx = -3\ln|x+1| + C_3\)Combine these results:\[-\ln|x| + 4\ln|x-2| - 3\ln|x+1| + C\]

Simplify the Final Answer

Combine the logarithmic expressions into a single expression:\[\ln \left(\frac{|x-2|^4}{|x| \cdot |x+1|^3}\right) + C\]Thus, the integral evaluates to \[\ln \left(\frac{|x-2|^4}{|x||x+1|^3}\right) + C\]

Key concepts

Integration Techniques

Integration techniques are essential tools in calculus for finding the antiderivative of functions. One widely used method when dealing with complex rational functions is **partial fraction decomposition**. This technique simplifies the integration process by breaking down a complicated rational function into simpler fractions that are easier to integrate. When the denominator can be factored into nonrepeated linear factors, partial fraction decomposition becomes straightforward.

To begin, break down the fraction's denominator into its prime factors. Once factored, you express the original function as a sum of simpler fractions. Each fraction has a single term from the factorization as its denominator. This allows each term to be integrated separately, often resulting in basic logarithmic functions. By splitting the integral into these manageable parts, integration becomes a neat and executable process, even for beginners.

• Factor the denominator first.
• Set up partial fractions.
• Integrate each fraction separately.

Using partial fraction decomposition is a neat trick to simplify difficult integrals, making them more approachable.

Rational Functions

Rational functions are a central part of algebra, comprising the quotient of two polynomials. In terms of integration, they often require special techniques due to their complexity. When dealing with partial fractions, the focus is on decomposing rational expressions to make them simpler to integrate.

A rational function is typically expressed in the form \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials. The degree of \(P(x)\) should be less than the degree of \(Q(x)\) for proper partial fraction decomposition. If it's not, polynomial long division might be necessary first.

Once you have this setup, you proceed to factor the denominator, if possible. The factored form helps in breaking down the original rational expression into simpler fractions using constants like \(A\), \(B\), and \(C\) determined through clearing the denominator and matching coefficients.

• Always ensure \(P(x)\) is of lesser degree than \(Q(x)\).
• Factor the denominator fully.
• Decompose into simpler terms for easier integration.

Rational functions, when simplified correctly, can reveal integrals that are straightforward to solve.

Linear Factors

Linear factors play an integral role in breaking down complex polynomial expressions in partial fractions. The term 'linear' refers to factors of the form \(ax + b\), where the variable is raised only to the first power.

In partial fraction decomposition, when a polynomial is factored into its linear components, it becomes a series of expressions, each associated with one of these factors. For each linear component from the denominator, you assign a corresponding term in the partial fraction, such as \(\frac{A}{x}\) for a factor \(x\).

Nonrepeated linear factors mean that each term in the denominator appears only once in the factored expression. This simplifies the process since each term in the resulting partial fraction decomposition is distinct and doesn't mix with others.

• Identify linear factors explicitly.
• Create a separate term for each linear factor in the decomposition.
• Individual linear factors make the integration straightforward.

Understanding linear factors helps in setting up partial fractions correctly, which directly aids in performing integrations smoothly.