Problem 30
Question
In Exercises \(29-34,\) perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral. $$ \int \frac{x^{4}}{x^{2}-1} d x $$
Step-by-Step Solution
Verified Answer
The integral evaluates to \( \frac{x^3}{3} + x + \frac{1}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C \).
1Step 1: Long Division
Divide the numerator \( x^4 \) by the denominator \( x^2 - 1 \). Divide \( x^4 \) by \( x^2 \) to get \( x^2 \). Multiply \( x^2 \) by \( x^2 - 1 \), resulting in \( x^4 - x^2 \). Subtract this from \( x^4 \) to get \( x^2 \). The result of the division is \( x^2 + \) remainder \( x^2 \). Thus, \( \frac{x^4}{x^2 - 1} = x^2 + \frac{x^2}{x^2 - 1} \).
2Step 2: Express as Sum of Partial Fractions
Focus on the proper fraction part \( \frac{x^2}{x^2 - 1} \). Rewrite \( x^2 \) as \((x^2 - 1) + 1\), so the fraction becomes: \( \frac{x^2-1+1}{x^2-1} = 1 + \frac{1}{x^2-1} \). Now, \( \int \frac{x^4}{x^2-1} dx = \int x^2 dx + \int \left(1 + \frac{1}{x^2-1}\right) dx \).
3Step 3: Evaluate the Integral of Whole Expression
Separate the integral into parts: \( \int x^2 dx + \int 1 dx + \int \frac{1}{x^2 - 1} dx \). Evaluate each part: \( \int x^2 dx = \frac{x^3}{3} + C_1 \), \( \int 1 dx = x + C_2 \). The integral of \( \frac{1}{x^2 - 1} \) requires partial fractions.
4Step 4: Further Decompose \( \int \frac{1}{x^2 - 1} dx \)
Factor the denominator: \( x^2 - 1 = (x-1)(x+1) \). Decompose \( \frac{1}{x^2 - 1} \) into partial fractions: \( \frac{1}{x^2 - 1} = \frac{A}{x-1} + \frac{B}{x+1} \). Solve for \( A \) and \( B \) to get \( A = \frac{1}{2} \, \) and \( B = -\frac{1}{2} \).
5Step 5: Integrate Partial Fractions
Now integrate each fraction: \( \int \frac{1}{x^2 - 1} dx = \int \left( \frac{1}{2(x-1)} - \frac{1}{2(x+1)} \right) dx \). This becomes \( \frac{1}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C_3 \).
6Step 6: Combine All Integrals Into Final Solution
Combine all the previously found integrals: \( \frac{x^3}{3} + x + \frac{1}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C \), where \( C = C_1 + C_2 + C_3 \).
Key Concepts
Long DivisionPartial FractionsIntegration TechniquesRational Functions
Long Division
Long division is a division technique used to simplify expressions where the degree of the numerator is greater than or equal to the degree of the denominator. This technique helps to divide polynomials more systematically, similar to dividing numbers.
To perform long division on the polynomial \( \frac{x^4}{x^2 - 1} \), start by dividing the leading term of the numerator by the leading term of the denominator. In this case, divide \( x^4 \) by \( x^2 \), resulting in \( x^2 \).
Next, multiply \( x^2 \) by the entire divisor \( x^2 - 1 \) to get \( x^4 - x^2 \). Subtract this product from the original polynomial \( x^4 \). This subtraction results in the remainder \( x^2 \).
Thus, after performing long division, the expression becomes \( x^2 + \frac{x^2}{x^2 - 1} \). This process simplifies the integration by reducing the degree of the polynomial, leaving an easier form to work with.
To perform long division on the polynomial \( \frac{x^4}{x^2 - 1} \), start by dividing the leading term of the numerator by the leading term of the denominator. In this case, divide \( x^4 \) by \( x^2 \), resulting in \( x^2 \).
Next, multiply \( x^2 \) by the entire divisor \( x^2 - 1 \) to get \( x^4 - x^2 \). Subtract this product from the original polynomial \( x^4 \). This subtraction results in the remainder \( x^2 \).
Thus, after performing long division, the expression becomes \( x^2 + \frac{x^2}{x^2 - 1} \). This process simplifies the integration by reducing the degree of the polynomial, leaving an easier form to work with.
Partial Fractions
Partial fractions are an algebraic technique used to decompose a complex rational expression into simpler fractions, making integration easier. This method is especially useful when dealing with rational functions where the numerator's degree is less than the denominator's degree.
For the fraction \( \frac{x^2}{x^2 - 1} \), let's rewrite it using a different approach. Note that \( x^2 = (x^2 - 1) + 1 \). So, the fraction can be expressed as \( \frac{x^2-1+1}{x^2-1} = 1 + \frac{1}{x^2-1} \).
This step transforms the rational function into a sum where traditional integration techniques can be applied. The key part now is to handle \( \frac{1}{x^2-1} \) using partial fraction decomposition. As the denominator factors into \((x-1)(x+1)\), set \( \frac{1}{x^2-1} = \frac{A}{x-1} + \frac{B}{x+1} \) and solve for constants \( A \) and \( B \).
For the fraction \( \frac{x^2}{x^2 - 1} \), let's rewrite it using a different approach. Note that \( x^2 = (x^2 - 1) + 1 \). So, the fraction can be expressed as \( \frac{x^2-1+1}{x^2-1} = 1 + \frac{1}{x^2-1} \).
This step transforms the rational function into a sum where traditional integration techniques can be applied. The key part now is to handle \( \frac{1}{x^2-1} \) using partial fraction decomposition. As the denominator factors into \((x-1)(x+1)\), set \( \frac{1}{x^2-1} = \frac{A}{x-1} + \frac{B}{x+1} \) and solve for constants \( A \) and \( B \).
Integration Techniques
Integration techniques refer to various methods used to find integrals, especially when direct integration is not straightforward. In processing the integral \( \int \frac{x^4}{x^2-1} \, dx \), different techniques have been applied to ease the evaluation process.
First, the expression was simplified through long division and partial fractions. The resulting integrals become \( \int x^2 \, dx \), \( \int 1 \, dx \), and \( \int \frac{1}{x^2-1} \, dx \).
For \( \int x^2 \, dx \) and \( \int 1 \, dx \), standard rules apply, leading to \( \frac{x^3}{3} + C_1 \) and \( x + C_2 \), respectively.
The integral \( \int \frac{1}{x^2-1} \, dx \), split using partial fractions, simplifies to two natural logarithmic integrals. Integrating each part yields \( \frac{1}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C_3 \). Combining all results gives the final solution.
First, the expression was simplified through long division and partial fractions. The resulting integrals become \( \int x^2 \, dx \), \( \int 1 \, dx \), and \( \int \frac{1}{x^2-1} \, dx \).
For \( \int x^2 \, dx \) and \( \int 1 \, dx \), standard rules apply, leading to \( \frac{x^3}{3} + C_1 \) and \( x + C_2 \), respectively.
The integral \( \int \frac{1}{x^2-1} \, dx \), split using partial fractions, simplifies to two natural logarithmic integrals. Integrating each part yields \( \frac{1}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C_3 \). Combining all results gives the final solution.
Rational Functions
Rational functions are ratios of two polynomials, expressed in the form \( \frac{P(x)}{Q(x)} \). They often appear in calculus problems, particularly in the context of integration. Understanding rational functions requires recognizing their forms and knowing how to simplify and decompose them effectively.
In the exercise, \( \frac{x^4}{x^2-1} \) is a rational function where the polynomial \( x^4 \) is the numerator, and \( x^2-1 \) is the denominator. The challenge with such functions during integration is reducing their complexity.
Different techniques like long division and partial fraction decomposition make these functions integrable by transforming them into simpler or standard forms. By working with these core concepts, tackling integrals of rational functions becomes manageable.
In the exercise, \( \frac{x^4}{x^2-1} \) is a rational function where the polynomial \( x^4 \) is the numerator, and \( x^2-1 \) is the denominator. The challenge with such functions during integration is reducing their complexity.
Different techniques like long division and partial fraction decomposition make these functions integrable by transforming them into simpler or standard forms. By working with these core concepts, tackling integrals of rational functions becomes manageable.
Other exercises in this chapter
Problem 30
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