Question

Use the method of partial fractions to evaluate the following integrals. \(\int \frac{3 x+4}{\left(x^{2}+4\right)(3-x)} d x\)

Short answer

The integral evaluates to \(-\frac{3}{2} \ln|x^2+4| - \frac{5}{2} \tan^{-1}(\frac{x}{2}) - 9 \ln|3-x| + C\).

Step-by-step solution

Decompose the Rational Function

The rational function \( \frac{3x + 4}{(x^2 + 4)(3 - x)} \) needs to be decomposed into partial fractions. This fraction can be written as the sum of the fractions \( \frac{Ax + B}{x^2 + 4} + \frac{C}{3-x} \). The common denominator is \((x^2 + 4)(3 - x)\).

Set Up the Partial Fraction Equation

Equate \( \frac{3x + 4}{(x^2 + 4)(3 - x)} = \frac{Ax + B}{x^2 + 4} + \frac{C}{3-x}\). Combine the partial fractions on the right with a common denominator to obtain: \( 3x + 4 = (Ax + B)(3 - x) + C(x^2 + 4) \).

Simplify and Combine Terms

Expand \((Ax + B)(3 - x) + C(x^2 + 4)\) to get \( (3A + C)x^2 + (-Ax + 3B)x + (3B + 4C) \). This should equal \(3x + 4\) on the left side.

Match Coefficients to Solve for Constants

Match the coefficients of \(x^2\), \(x\), and the constant terms on both sides: \[3A + C = 0\]\[-A = 3\]\[3B + 4C = 4\]From \(-A = 3\), solve to get \(A = -3\). Substitute \(A = -3\) into the first equation: \(3(-3) + C = 0\) gives \(C=9\). Substitute \(C = 9\) into the third equation: \(3B + 4(9) = 4\) which gives \(B = -10\).

Rewrite the Integral Using Partial Fractions

The integral becomes \( \int \left(\frac{-3x - 10}{x^2 + 4} + \frac{9}{3 - x}\right) \, dx \).

Integrate Each Term Separately

Integrate each part: 1. \( \int \frac{-3x - 10}{x^2 + 4} \, dx = \int \left( \frac{-3x}{x^2 + 4} + \frac{-10}{x^2 + 4} \right) \, dx \) - Use substitution methods for each.2. \( \int \frac{9}{3-x} \, dx = -9 \ln|3-x| + C_2 \)Integrating: - For \( \int \frac{-3x}{x^2+4} \, dx\), use substitution \( u = x^2 + 4, \ du = 2x \, dx \).- For \( \int \frac{-10}{x^2+4} \), note the integral is related to \( \tan^{-1}\).

Evaluate the Integrals

Solving the integrals: - For \( \int \frac{-3x}{x^2+4} \, dx \), use substitution to find \( -\frac{3}{2} \ln|x^2+4| \).- For \( \int \frac{-10}{x^2+4} \, dx \), recognize the form \( \frac{1}{a}\tan^{-1}(\frac{x}{a}) \) with \( a=2 \), leading to \( -\frac{5}{2} \tan^{-1}(\frac{x}{2}) \).- Combine results from step 6 and 7: \(-\frac{3}{2} \ln|x^2+4| - \frac{5}{2} \tan^{-1}(\frac{x}{2}) - 9 \ln|3-x| + C\)

Key concepts

Integral Calculus

Integral calculus is a fundamental branch of mathematics that focuses on the concept of integration, which is essentially the reverse process of differentiation. It originates from the idea of finding the area under a curve or between curves, which is applied in various scientific and engineering fields.

There are two primary operations in integral calculus:

• Definite integrals, which calculate the exact area under a curve between two points.
• Indefinite integrals, which represent a family of functions and include a constant of integration, often denoted by the letter 'C'.

Integration has numerous applications, such as solving problems related to:

• Finding areas and volumes of geometric shapes.
• Calculating distances and velocities in physics.
• Evaluating economic models in finance.

Understanding integral calculus is critical as it provides tools for analyzing and interpreting various natural phenomena and systems. In this context, integration plays a vital role in evaluating complex expressions like rational functions.

Rational Functions

Rational functions are fractions composed of polynomials and are commonly seen in mathematical analyses. They take the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomial functions.

Rational functions can exhibit a variety of behaviors, based on the degrees of the numerator and the denominator:

• If the degree of \(P(x)\) is less than that of \(Q(x)\), the rational function is generally proper.
• If the degree of \(P(x)\) is equal to or greater than that of \(Q(x)\), the function is considered improper and might require polynomial long division for simplification.

When dealing with complex integrals involving rational functions, decomposing them into simpler partial fractions helps in integration. This technique is particularly useful when the denominator can be factored into simpler linear or quadratic terms, as seen in our exercise.

Integration Techniques

Integration can often be challenging due to the complexity of the functions involved. Luckily, there are several techniques to simplify and solve integrals. Two key methods are particularly relevant when handling rational functions:

Partial Fraction Decomposition: This technique involves breaking down a complex fraction into a sum of simpler fractions, which makes integration easier.

• By expressing the integrand as a sum of simpler parts, each term becomes more manageable for integration.
• This method requires that the function be properly set up, where the degree of the numerator is less than that of the denominator.

Substitution Method: This method involves substituting a part of the integral with a new variable to simplify the integrand and make the integral more manageable.

• It is typically used when dealing with compositions of functions, where a direct integration isn't straightforward.
• For example, when facing an integral of a function like \(\frac{-3x}{x^2+4}\), substitution allows for an easier computation process.

Both techniques require practice to master and apply effectively in various integration scenarios.

Substitution Method

The substitution method is a powerful technique used to simplify complex integrals, particularly when dealing with complicated expressions within the integrand. The idea is to identify a part of the integrand as a new variable, making the integral easier to handle.

**Steps to Apply Substitution Method:**

• Choose a substitution variable that will simplify the integrand. Often, this is a term that appears under a radical or inside a trigonometric or logarithmic function.
• Express the original variable and differential in terms of the new variable. This typically requires finding \(du = g(x) \, dx\), where \(g(x)\) is the derivative of the substitution variable.
• Rewrite the integral in terms of the new variable and solve it.
• After integrating, substitute back the original variable.

For the exercise, when tackling an integral like \(\int \frac{-3x}{x^2+4} \, dx\), we use \(u = x^2+4\), leading to an easier expression in terms of \(u\). This approach transforms complex integrals into more straightforward problems, allowing us to solve a wide range of challenging calculus questions.