Problem 7 - Essential Calculus

Question

Use partial fractions to find the integral. $$ \int \frac{5-x}{2 x^{2}+x-1} d x $$

Step-by-Step Solution

Answer

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

Step 1: Factorize the Denominator

Let's start by factorizing the denominator \(2x^{2} + x - 1\). By inspection, we can write this as \((2x - 1)(x + 1)\). After the factorization, our integral becomes \(\int \frac{5-x}{(2x - 1)(x + 1)} dx\)

Step 2: Partition into Partial Fractions

Write the fraction on the right hand side in the form of two separate fractions. To do this, express \(\frac{5 - x}{(2x - 1)(x + 1)}\) as \(\frac{A}{2x-1} + \frac{B}{x+1}\). Multiply both sides by the denominator \((2x-1)(x+1)\) which gives us \(5 - x = A(x + 1) + B(2x - 1)\). Set x=-1, A=(4). Set x=1/2, B=(-3). Therefore the our integral becomes two separate integrals. That is, \(\int \frac{4}{2x-1} dx - \int \frac{3}{x+1} dx\).

Step 3: Perform Integration

Now integrate each part separately. The integral of \(\frac{4}{2x-1}\) is \(2 \ln|2x-1|\) and the integral of \(\frac{3}{x+1}\) is \(3 \ln|x+1|\). Thus, the integration gives \(2 \ln|2x-1| - 3 \ln|x+1| + C\).

Step 4: Simplify Result

For simplification, write the result in terms of natural logarithmic laws. That is, \(2 \ln|2x-1| - 3 \ln|x+1| + C = \ln|(2x-1)^2|- \ln|(x+1)^3| + C = \ln| \frac{(2x-1)^2}{(x+1)^3}| + C\), where \(C\) is the constant of integration.

Key Concepts

Integration Techniques

Integration techniques are essential tools for solving complex integrals. When confronted with an integral involving a rational function, strategic methods can simplify the process. Partial fractions are often a handy technique in these cases. They allow us to decompose a complex fraction into simpler parts, which are then easier to integrate.
To employ the partial fraction method, it is important to understand how to write a complicated fraction as a sum of simpler fractions. This involves breaking down the denominator into manageable factors.

• Identify the factors of the denominator.
• Express the complex fraction using unknown numerators over each factor.
• Solve for these unknowns to simplify the problem.

By doing so, integrations are simplified into basic forms like logarithmic or polynomial forms. Such techniques can turn otherwise daunting problems into solvable steps.

Rational Functions

Rational functions are ratios of two polynomials, where the degree of the numerator is less than or equal to the degree of the denominator. They play a big role in calculus, especially when dealing with integration.
In the provided problem, the rational function is \( \frac{5-x}{2x^2 + x - 1} \). Before applying any integration method, a crucial step is to factor the polynomial denominator. For example, in the given exercise, the polynomial \(2x^2 + x - 1\) was factorized as \((2x-1)(x+1)\).

• Ensure that the polynomial is fully factorized.
• Check for any simplifiable terms between the numerator and the denominator.
• Use these factors in partial fraction decomposition.

These actions make it easier to tackle the integral, as each fraction term can often be integrated using straightforward methods like logarithmic integration.

Logarithmic Integration

Logarithmic integration is a technique used when integrating fractions where the denominator can be expressed as a simple linear term. This method relates directly to partial fractions when breaking down complex rational functions.
Once the rational function is decomposed into simpler fractions, each part can often be integrated using basic logarithmic rules. For example, integrating \( \frac{1}{ax+b} \) results in \( \ln|ax+b| \).
This kind of integration was used in the solution to handle each of the partial fractions separately:

• The integral of \( \frac{4}{2x-1} \) resulted in \( 2 \ln|2x-1| \).
• The integral of \( \frac{3}{x+1} \) resulted in \( 3 \ln|x+1| \).

In these cases, the coefficients outside the fractions multiply the logarithm. Always remember to apply logarithmic rules when combining or separating the results after integration. This ensures simplicity and clarity in the final answer.