Problem 38 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The result of the definite integral is -7/4

Step 1 - Divide The Integral

We divide the integral into two separate parts: \[ \int_{0}^{1} \frac{x^{3}}{x^{2}-4} dx - \int_{0}^{1} \frac{1}{x^{2}-4} dx \]

Step 2 - Simplify The Integrals

The first integral can be simplified using the power rule: \[ \int_{0}^{1} \frac{x^{3}}{x^{2}-4} dx = [\frac{x^2}{2}]-[-1*(\frac{4}{x})] \Big|_0^1 = \frac{1}{2} - 4 \] The second integral can be calculated using the formula \(\int \frac{1}{x^2+a^2} dx = \frac{1}{a} * tan^{-1}(\frac{x}{a})\). Here \(a^2 = 4 = 2^2\). Therefore, \[\int_{0}^{1} \frac{1}{x^{2}-4} dx = \frac{1}{2*tan^{-1}(1/2)} \Big|_0^1 = \frac{1}{4}\]

Step 3 - Subtract The Results

Now subtract the two results to get the final answer: \((\frac{1}{2} - 4) - \frac{1}{4}= -\frac{7}{4}\)

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics that deals with integrals, which are essentially the reverse process of differentiation. It involves finding the area under a curve on a graph, which is symbolized by the integral sign \( \int \).
Definite integrals, specifically, calculate the area under a curve between two points, giving a numerical value that represents this area. Unlike indefinite integrals, they do not include a constant of integration.
This particular problem asks us to evaluate the definite integral \( \int_{0}^{1} \frac{x^{3}-1}{x^{2}-4} dx \), which means finding the total area under the curve of the function \( \frac{x^{3}-1}{x^{2}-4} \) from \( x = 0 \) to \( x = 1 \).

Integration Techniques

When solving integrals, especially complex ones involving fractions like \( \frac{x^{3}-1}{x^{2}-4} \), we need various integration techniques.
These techniques help simplify the integrals or make them integrable. Some common techniques include:

• Simplification: Breaking down complex integrals into more manageable parts.
• Substitution: Replacing variables or expressions to simplify an integral.
• Partial Fraction Decomposition: Used for rational functions, separating them into simpler fractions.
• Trigonometric Substitution: Applying trigonometric identities to simplify integrals involving roots or powers.

In this exercise, we divided the integral into two parts: \( \int_{0}^{1} \frac{x^{3}}{x^{2}-4} dx \) and \( \int_{0}^{1} \frac{1}{x^{2}-4} dx \) to tackle them separately.

Power Rule

The power rule in integration is one of the simplest and most frequently used techniques. It involves integrating functions of the form \( x^n \).
The power rule states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration for indefinite integrals.
In the example provided, the power rule helped simplify the first integral, leading to \( \int_{0}^{1} \frac{x^{3}}{x^{2}-4} dx \). Calculations yielded \( \frac{1}{2} - 4 \) using simplification and the power rule. While the formula might seem complicated, it essentially reduces the work by handling powers of \( x \) directly.

Trigonometric Substitution

Trigonometric substitution is an insightful technique when faced with integrals that include expressions like \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), and \( \sqrt{x^2 - a^2} \).
Though it appears complex, it uses trigonometric identities to convert these into integrable forms. This method is particularly useful when the integral involves quadratic expressions.
In the solution, we relied on a related method, involving an alternative form from trigonometric integrals: \( \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C \).
With \( a = 2 \), the integral \( \int_{0}^{1} \frac{1}{x^{2}-4} dx \) transformed into a manageable form, leading us to find \( \frac{1}{4} \) for its integral part.
This exemplifies how substitution techniques can streamline solving complex integrals.