Problem 23 - Calculus Early Transcendentals

Question

Evaluate the following integrals. $$\int \frac{x^{2}+12 x-4}{x^{3}-4 x} d x$$

Step-by-Step Solution

Answer

Answer: The integral is $$\ln |x| + \frac{2}{3} \ln |x^3 - 4| + C$$.

Step 1: Rewrite the integrand using partial fraction decomposition

First, we can try to decompose the integrand into partial fractions. Let's rewrite the integrand in terms of partial fraction decomposition: $$\frac{x^{2}+12x-4}{x^3-4x} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3-4}$$ To find the constants A, B, and C, we need to clear denominators: x^2 + 12x - 4 = A(x^2)(x-4) + Bx(x-4) + Cx(x^2)

Step 2: Solve for the constants A, B, and C

To solve for A, B, and C, we substitute convenient values for x to remove other terms: - For A: let x = 0, then we have -4 = -4A ==> A = 1 - For B: differentiate both sides of the equation: 2x + 12 = 8A + (1-4x)B + x^2C Substitute x = 0, we have 12 = 8(1) + 0B + 0C ==> B = 0 - For C: differentiate both sides of the equation one more time: 2 = -4B + 2xC Substitute x = 0, we have 2 = -4(0) + C ==> C = 2

Step 3: Rewrite integrand with found constants

Now that we have the values for A, B, and C, rewrite the integrand: $$\frac{x^{2}+12x-4}{x^3-4x} = \frac{1}{x} + \frac{2}{x^3-4}$$

Step 4: Integrate term by term

Next, integrate each term separately: $$\int \frac{1}{x} dx = \ln |x| + C_1$$ $$\int \frac{2}{x^3-4} dx = \frac{2}{3} \int \frac{1}{u} du$$ (Here, we made a substitution u = x^3 - 4, so du = 3x^2 dx)

Step 5: Combine the individual integrals

Now, we combine the two individual integrals to get the final result: $$\int \frac{x^{2}+12 x-4}{x^{3}-4 x} d x = \ln |x| + \frac{2}{3} \ln |x^3 - 4| + C$$

Key Concepts

Integration Techniques

Integration techniques are a set of strategic methods used to simplify and solve complex integrals. Often, the direct integration of complex expressions is not feasible. So, we use various techniques to break them down into simpler parts. Some common integration techniques include:

• Partial Fraction Decomposition: Breaking down a complex fraction into simpler fractions to make the integration process easier.
• Substitution Method: Changing the variable of integration to simplify the integral.
• Integration by Parts: A technique that removes the product by transforming it into a simpler integral.

In the given exercise, we utilized partial fraction decomposition to rewrite the rational function into simpler terms. The original function \(rac{x^{2}+12x-4}{x^3-4x}\) was decomposed to \(rac{1}{x} + \frac{2}{x^3-4}\), making it easier to apply basic integration formulas. This technique becomes particularly useful when dealing with rational expressions, providing a clearer path toward integration.

Definite and Indefinite Integrals

When we talk about integrals, we generally refer to two types: definite and indefinite integrals. Definite integrals are used to calculate the area under a curve within a specific interval, while indefinite integrals are used to find the antiderivative of a function.

In our exercise, we dealt with an indefinite integral, which means the function was integrated without specifying any limits of integration. An indefinite integral is expressed as:\[ \int f(x) \, dx = F(x) + C \]where \(F(x)\) is the antiderivative of \(f(x)\) and \(C\) is the constant of integration.

Here, after decomposing the expression into simpler fractions, we found the antiderivatives of each term. We integrated term by term, resulting in: \[\ln |x| + \frac{2}{3} \ln |x^3 - 4| + C\]This encapsulates the complete antiderivative of the original function.

Substitution Method

The substitution method is a powerful technique used to simplify integrals, especially those involving composite functions or where a function can be represented in terms of another variable. This method involves introducing a new variable, \(u\), to replace an expression in the integrand.

Here's how substitution can simplify integration:

• Identify a part of the integrand that can be replaced with the new variable \(u\).
• Compute the differential \(du\) in terms of \(dx\).
• Re-express the integral entirely in terms of \(u\) and \(du\).

In our exercise, substitution was necessary while integrating the term \(rac{2}{x^3-4}\). We set \(u = x^3 - 4\), so \(du = 3x^2 \, dx\). This transformation allowed the integral to be split into a simpler form, using the natural logarithm function for the final evaluation.

Through substitution, complex integrals can often be resolved into forms amenable to straightforward integration techniques, easing the overall integration process.