Problem 67
Question
Calculate the given integral. $$ \int \frac{x^{2}-6 x+8}{x^{2}+4} d x $$
Step-by-Step Solution
Verified Answer
The integral is \(x - 3 \ln|x^2 + 4| + 2 \tan^{-1} \left( \frac{x}{2} \right) + C\).
1Step 1: Identify the Type of Integral
This integral is a rational function, which is a fraction where the numerator and the denominator are polynomials. We can solve this by performing polynomial long division since the degree of the polynomial in the numerator is equal to the degree of the polynomial in the denominator.
2Step 2: Perform Polynomial Long Division
Divide the numerator by the denominator: - Divide the leading term of the numerator, which is \(x^2\), by the leading term of the denominator, \(x^2\), to get \(1\).- Multiply \(1\) by \(x^2 + 4\) to get \(x^2 + 4\).- Subtract \(x^2 + 4\) from \(x^2 - 6x + 8\) which results in \(-6x + 4\).This gives us \(1 + \frac{-6x+4}{x^2+4}\).
3Step 3: Separate the Integral
The integral can now be rewritten as two separate integrals:\[\int 1 \, dx + \int \frac{-6x + 4}{x^2 + 4} \, dx\]The first integral, \(\int 1 \, dx\), is straightforward and equals \(x\).
4Step 4: Decompose and Solve the Second Integral
The second integral is \(\int \frac{-6x + 4}{x^2 + 4} \, dx\). To solve this, decompose \(-6x + 4\) into parts that fit the form of known derivatives.Rewrite the fraction as:\[\int \left( -6 \cdot \frac{x}{x^2 + 4} + \frac{4}{x^2 + 4} \right) \, dx\]
5Step 5: Solve Each Sub-Integral
For \(-6 \cdot \int \frac{x}{x^2 + 4} \, dx\), use the substitution \(u = x^2 + 4\), which means \(du = 2x \, dx\) or \(x \, dx = \frac{1}{2} \, du\). This gives:\[-3 \cdot \int \frac{1}{u} \, du = -3 \ln|u| + C = -3 \ln|x^2 + 4| + C\]For \(\int \frac{4}{x^2 + 4} \, dx\), recognize it as a standard integral:\[4 \cdot \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right) = 2 \tan^{-1}\left(\frac{x}{2}\right)\]
6Step 6: Write Down the Final Result
Combine all the results from previous steps:\[x - 3 \ln|x^2 + 4| + 2 \tan^{-1}\left(\frac{x}{2}\right) + C\]where \(C\) is the constant of integration.
Key Concepts
Polynomial Long DivisionSubstitution MethodIntegral CalculusPartial Fraction Decomposition
Polynomial Long Division
Polynomial long division is similar to standard long division that you may know from basic arithmetic. Here, it is specifically used to divide polynomials. This technique helps in simplifying rational expressions where the numerator has a degree greater than or equal to the denominator.
To begin, you divide the leading term of the numerator by the leading term of the denominator. In this exercise, both are \(x^2\), so dividing gives \(1\). Next, multiply this result by the entire denominator, which results in \(x^2+4\). Subtract this from the original numerator \(x^2-6x+8\), resulting in the remainder \(-6x+4\).
Finally, express the original fraction as the sum of this quotient \(+\) the fraction with the remainder as its numerator over the original denominator. Here, it simplifies our integral to: \(1 + \frac{-6x+4}{x^2+4}\).
Using polynomial long division makes complex rational functions much easier to handle and analyze. It helps particularly when integrating, where simplifying the integrand can significantly ease the calculation.
To begin, you divide the leading term of the numerator by the leading term of the denominator. In this exercise, both are \(x^2\), so dividing gives \(1\). Next, multiply this result by the entire denominator, which results in \(x^2+4\). Subtract this from the original numerator \(x^2-6x+8\), resulting in the remainder \(-6x+4\).
Finally, express the original fraction as the sum of this quotient \(+\) the fraction with the remainder as its numerator over the original denominator. Here, it simplifies our integral to: \(1 + \frac{-6x+4}{x^2+4}\).
Using polynomial long division makes complex rational functions much easier to handle and analyze. It helps particularly when integrating, where simplifying the integrand can significantly ease the calculation.
Substitution Method
The substitution method in calculus involves replacing a complicated part of an integrand with a single variable to simplify the integral. This technique is especially useful when dealing with integrals involving composite functions or when the denominator is a derivative of the numerator.
For \(-6 \cdot \int \frac{x}{x^2 + 4} \, dx\):
For \(-6 \cdot \int \frac{x}{x^2 + 4} \, dx\):
- We let \(u = x^2 + 4\), which simplifies the integrand since \(du = 2x \, dx\) or equivalently \(x \, dx = \frac{1}{2} \, du\).
- This transforms our integral into \(-3 \cdot \int \frac{1}{u} \, du\), a much simpler integral to solve, which results in \(-3 \ln|u| = -3 \ln|x^2 + 4|\).
Integral Calculus
Integral calculus involves finding the integral or antiderivative of a function. This concept is fundamental in finding areas under curves and in various applications such as physics and engineering. In our exercise, we are finding the integral of a rational function.
To perform integration, you may sometimes need to split the integral into more manageable parts, as done in steps 3 and 4. The function was split into two distinct integrals: \( \int 1 \, dx = x \) and \( \int \frac{-6x+4}{x^2+4} \, dx \).
Integral calculus can involve different techniques depending on the function being integrated. Techniques like substitution can simplify parts of the integral, while polynomial long division assists with rational functions. Mastering these techniques can help tackle a wide variety of problems in calculus.
To perform integration, you may sometimes need to split the integral into more manageable parts, as done in steps 3 and 4. The function was split into two distinct integrals: \( \int 1 \, dx = x \) and \( \int \frac{-6x+4}{x^2+4} \, dx \).
Integral calculus can involve different techniques depending on the function being integrated. Techniques like substitution can simplify parts of the integral, while polynomial long division assists with rational functions. Mastering these techniques can help tackle a wide variety of problems in calculus.
Partial Fraction Decomposition
Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions that are more straightforward to integrate. However, in our specific exercise, instead of traditional partial fraction decomposition, we decomposed the integral into separate terms using our earlier work with polynomial long division.
Traditional partial fraction decomposition would typically apply when the degree of the numerator is less than the degree of the denominator. While our integral did not directly use partial fraction decomposition, understanding this concept remains crucial for more complex integrals, particularly when the denominator can be factored. It allows us to transform a complicated rational function into a sum of simpler fractions, each of which can be integrated separately.
Traditional partial fraction decomposition would typically apply when the degree of the numerator is less than the degree of the denominator. While our integral did not directly use partial fraction decomposition, understanding this concept remains crucial for more complex integrals, particularly when the denominator can be factored. It allows us to transform a complicated rational function into a sum of simpler fractions, each of which can be integrated separately.
Other exercises in this chapter
Problem 67
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