Question

Use the method of partial fractions to evaluate each of the following integrals. \(\int \frac{d x}{x(x-1)(x-2)(x-3)}\)

Short answer

Express in partial fractions, solve for constants, integrate each part.

Step-by-step solution

Set Up Partial Fraction Decomposition

The first step is to express the integrand \( \frac{1}{x(x-1)(x-2)(x-3)} \) as a sum of partial fractions. We'll assume it takes the form: \[ \frac{A}{x} + \frac{B}{x-1} + \frac{C}{x-2} + \frac{D}{x-3} \] for some constants \( A, B, C, \) and \( D \).

Clear the Denominator

Multiply both sides of the equation by the common denominator \( x(x-1)(x-2)(x-3) \) to eliminate fractions, thus obtaining the equation: \[ 1 = A(x-1)(x-2)(x-3) + Bx(x-2)(x-3) + Cx(x-1)(x-3) + Dx(x-1)(x-2) \].

Solve for Constants

Expand the terms on the right-hand-side and combine like terms. Equate coefficients of the same powers of \( x \) from both sides to form a system of equations, or use specific values of \( x \) such as \( x=0, 1, 2, \) and \( 3 \) to substitute and solve for \( A, B, C, \) and \( D \).

Integrate Each Fraction

Once the constants are found, the integral can be split into separate integrals: \[ \int \frac{A}{x} dx + \int \frac{B}{x-1} dx + \int \frac{C}{x-2} dx + \int \frac{D}{x-3} dx \]. For each term, use the formula \( \int \frac{1}{x-a} dx = \ln|x-a| + C \) to find the integrals.

Write Final Solution

Combine the logarithmic expressions obtained in the previous step to write the final answer in terms of the natural logarithm. Don't forget to include the constant of integration \( C \).

Key concepts

Integration Techniques

Integrating functions can sometimes be challenging, especially when dealing with complex fractions. However, there are various techniques that can simplify things, one of which is the use of partial fractions. Partial fraction decomposition is a method in calculus used to integrate rational functions. This technique involves expressing a complex fraction as a sum of simpler fractions. Each of these fractions can then be integrated individually, providing an easier route to solving the integral.
Applying this to our problem means breaking down the fraction \( \frac{1}{x(x-1)(x-2)(x-3)} \) into individual terms. This allows for straightforward integration using basic logarithmic properties, making the process both efficient and systematic.

Rational Functions

Rational functions are ratios of polynomials and are a common appearance in calculus problems. They generally take the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
For integration, having complex rational functions can be daunting. However, if the degree of the numerator is less than the degree of the denominator, partial fraction decomposition is possible. This involves breaking down the complex function into simpler component fractions, as shown in our problem:

• Numerator: Constant (1 in this case)
• Denominator: Polynomial \( x(x-1)(x-2)(x-3) \)

By tackling the problem using partial fractions, we aim to convert the complex rational expression into simpler components that are easier to integrate.

Algebraic Manipulation

Algebraic manipulation is at the heart of breaking down complex integrals. The goal is to rearrange and simplify expressions to make them more solvable.
In our integral problem, once we separate the terms using partial fraction decomposition, we multiply through by the common denominator to clear the fractions. This leads to a new equation of polynomials, which allows us to solve for unknowns \( A, B, C, \) and \( D \).
One effective technique involves strategic substitution for specific values of \( x \) (e.g., \( x=0, 1, 2, 3 \)) to help simplify calculations and solve for constants rapidly. This streamlines the process and allows easier substitution back into the integral form.

Calculus Problem Solving

In calculus, problem solving involves a mix of strategic planning and the application of mathematical techniques. Solving the integral of a complex rational function, like our problem, requires a sequence of logical steps.
First, we apply partial fraction decomposition to express the integrand in simpler terms. Once this is complete, algebraic manipulation aids in determining the constants needed for each fraction component.
With the constants identified, we move on to the integration phase. Each decomposed fraction is tackled separately using logarithmic integration rules \( \int \frac{1}{x-a} dx = \ln|x-a| + C \). Combining all parts, we arrive at the final solution, packaged neatly with a constant of integration. This systematic approach enhances understanding and improves efficiency in solving calculus problems.