Problem 25 - Calculus: Early Transcendental Functions

Question

Evaluate the integral. $$\int_{0}^{1} \frac{x^{2}}{x^{3}-4} d x$$

Step-by-Step Solution

Answer

The evaluated result of the integral \( \int_{0}^{1} \frac{x^{2}}{x^{3}-4} d x \) depends on the calculated values of A, B, and C from partial fractions decomposition and the evaluation of the resultant simpler integrals.

Step 1: Perform Partial Fractions Decomposition

Express the integrand \(\frac{x^{2}}{x^{3}-4}\) in terms of its partial fractions. To do this, first factorize the denominator \(x^{3}-4\) as \((x-2)(x^{2}+2x+2)\). Then write \(\frac{x^{2}}{(x-2)(x^{2}+2x+2)} = \frac{A}{x-2} + \frac{Bx+C}{x^{2}+2x+2}\). Identify the values of A, B, and C by equating coefficients.

Step 2: Integrate the Partial Fractions

Integrate each partial fraction expression obtained from step 1. Note that the second term is a quadratic expression hence completing the square could be used to find an antiderivative. So write \( \int_{0}^{1} \frac{A}{x-2} dx + \int_{0}^{1} \frac{(Bx+C)}{x^{2}+2x+2} dx \). Calculate each integral separately.

Step 3: Evaluate the Definite Integrals

Substitute the values of the limits of 0 and 1 into the antiderivatives obtained in step 2. Subtract the value at the lower limit from the value at the upper limit for each and add up all the results.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition helps to break down complex rational expressions into simpler, more manageable parts. This process involves expressing a rational function as a sum of simpler fractions, which is particularly useful in integrating rational functions. To decompose the fraction \(\frac{x^{2}}{x^{3}-4}\), the first step is to factor the denominator. The expression \(x^{3}-4\) can be factored as \((x-2)(x^{2}+2x+2)\).
Next, we express the original fraction as a sum of partial fractions:

• \(\frac{A}{x-2}\)
• \(\frac{Bx+C}{x^{2}+2x+2}\)

This gives the equation:\[\frac{x^{2}}{(x-2)(x^{2}+2x+2)} = \frac{A}{x-2} + \frac{Bx+C}{x^{2}+2x+2}\] To find \(A\), \(B\), and \(C\), multiply through by the denominator \((x-2)(x^{2}+2x+2)\) and solve for the coefficients by matching terms. This results in a system of equations that can be solved to determine \(A\), \(B\), and \(C\). These coefficients are vital to simplifying the integration process.

Definite Integrals

A definite integral is an integral that computes the accumulation of quantities over a specified interval. When evaluating the definite integral \(\int_{a}^{b} f(x) \, dx\), it's crucial to determine the antiderivative first, then apply the fundamental theorem of calculus.
In this exercise, we aim to evaluate the definite integral \(\int_{0}^{1} \frac{x^{2}}{x^{3}-4} dx\) by first using partial fraction decomposition to split the integrand into simpler fractions. Once decomposed, we find the antiderivative for each separate term over the interval [0, 1].
This involves:

• Finding the antiderivative of each partial fraction.
• Substituting the upper and lower limits into the antiderivative expression.
• Calculating the difference between these values.

Together, these steps yield the value of the definite integral, representing the total accumulation of \(f(x)\) from 0 to 1.

Antiderivatives

Antiderivatives are fundamental in calculus, particularly for solving integrals. Essentially, an antiderivative of a function \(f(x)\) is another function \(F(x)\), whose derivative yields \(f(x)\). Finding antiderivatives is crucial in the integration process, especially for evaluating definite integrals.
In our exercise, after decomposing the original function into partial fractions, we resolve each simpler fraction by finding its antiderivative. Some common methods to find antiderivatives include:

• Basic power rule for polynomials.
• Substitution for more complex expressions like \(\int \frac{1}{u} du\), which results in \(\ln|u|+C\).

For terms that result in a quadratic denominator, completing the square may be useful in finding its antiderivative effectively.

Completing the Square

Completing the square is a technique used to simplify expressions, particularly quadratics, and is very beneficial in integration. This technique involves rewriting a quadratic expression in the form \((x+p)^{2}+q\), making it easier to integrate, especially when finding the antiderivative.
For the quadratic denominator \(x^{2}+2x+2\) in our fraction, completing the square helps rewrite it in a more workable form. Here's how it's done:

• Start with \(x^2 + 2x + 2\).
• Rearrange it as \((x+1)^2 + 1\) by adding and subtracting 1 from inside and outside of the square.

This conversion simplifies the process of integration, particularly for expressions resembling a natural logarithm or an arctan function. Through completing the square, we reduce complex quadratic terms into forms that are simpler to integrate.