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Chapter 8: Problem 58

$$\text {Evaluate the following integrals.}$$ $$\int \frac{x^{4}+1}{x^{3}+9 x} d x$$

Short Answer

Expert verified
To evaluate the integral of the rational function \(\int \frac{x^4+1}{x^3+9x} dx\), we started by performing long division, breaking the integrand into simpler terms. We rewrote the integrand as \(x+\frac{1-9x^2}{x^3+9x}\). After factoring the denominator and using partial fraction decomposition, we obtained \(\frac{1}{9x}- \frac{9x}{x^2+9}\). Next, we integrated each term separately and combined the results. The final answer is: $$\int \frac{x^{4}+1}{x^{3}+9 x} d x = \frac{1}{2}x^2 + \frac{1}{9}\ln|x| - \frac{9}{2}\ln|x^2+9| + C$$, where C is the constant of integration.

Step by step solution

01

Divide \(x^4+1\) by \(x^3+9x\) to get \(\frac{x^4+1}{x^3+9x}=x+\frac{1-9x^2}{x^3+9x}\). Now we have a new integral to solve: $$\int x+\frac{1-9x^2}{x^3+9x}dx$$ #Step 2: Factor the denominator# Now, we want to factor the denominator of the fraction in the new integral.

Factor the denominator of the fraction: \(x^3+9x = x(x^2+9)\). #Step 3: Use partial fraction decomposition# Next, we will decompose the fraction \(\frac{1-9x^2}{x(x^2+9)}\) into partial fractions.
02

Let \(\frac{1-9x^2}{x(x^2+9)} = \frac{A}{x}+\frac{Bx+C}{x^2+9}\). Multiply both sides of this equation by the original denominator, \(x(x^2+9)\), to get rid of the denominators. We have: \(1-9x^2 = A(x^2+9) + x(Bx+C)\). #Step 4: Solve for A, B, and C# Now we will determine the values for A, B, and C.

To solve for A, set \(x=0\), so the equation becomes: \(1 = 9A \Rightarrow A = \frac{1}{9}\). To find B and C, we can equate the coefficients of the powers of x on both sides of the equation. This gives us the following system of equations: - \(0B + 1C = 0 \Rightarrow C = 0\) - \(1B + 0A = -9 \Rightarrow B = -9\) Now, we can rewrite the fraction as: \(\frac{1}{9x} - \frac{9x}{x^2+9}\). So, our new integral to solve is: $$\int \left(x+\frac{1}{9x}- \frac{9x}{x^2+9}\right)dx$$ #Step 5: Integrate each term separately# Now we will integrate each term in the expression separately.
03

- For the first term, \(\int x dx = \frac{1}{2}x^2+C_1\). - For the second term, \(\int \frac{1}{9x} dx = \frac{1}{9} \int \frac{1}{x} dx = \frac{1}{9} \ln|x|+C_2\). - For the third term, \(\int \frac{-9x}{x^2+9} dx = -9\int \frac{x}{x^2+9} dx\). To solve this integral, we can use the substitution method. Let \(u=x^2+9\). Then \(du=2x dx\), or \(\frac{1}{2}du = x dx\). Substitute and solve the integral: $$-9\int \frac{x}{x^2+9} dx= -\frac{9}{2}\int \frac{1}{u} du = -\frac{9}{2}\ln |u|+C_3 = -\frac{9}{2}\ln|x^2+9|+C_3$$. #Step 6: Combine the results# Finally, we combine the results of all the individual integrals to obtain the final answer.

$$\int \frac{x^{4}+1}{x^{3}+9 x} d x = \frac{1}{2}x^2 + \frac{1}{9}\ln|x| - \frac{9}{2}\ln|x^2+9| + C$$, where \(C = C_1+C_2+C_3\) is the constant of integration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Fraction Decomposition
Understanding partial fraction decomposition is essential when dealing with integrals of rational functions, particularly when the degree of the numerator is equal to or greater than the degree of the denominator. The goal of this technique is to break down a complex fraction into simpler fractions that are easier to integrate.

In our exercise, the fraction \(\frac{1-9x^2}{x(x^2+9)}\) needed to be decomposed. We searched for constants \(A\), \(B\), and \(C\) such that \(\frac{1-9x^2}{x(x^2+9)} = \frac{A}{x}+\frac{Bx+C}{x^2+9}\). By clearing the denominators, we create an equation that allows us to solve for these constants. This step is crucial because it transforms the original integral into a sum of simpler integrals that are more straightforward to solve.

To optimize learning, while dealing with partial fractions, it is suggested that the student should:
  • First, ensure the degree of the numerator is less than that of the denominator; if not, use polynomial long division first.
  • Familiarize themselves with various cases of partial fraction decomposition and know when to apply each.
  • Remember that the goal is to simplify complex expressions, making them more manageable.
Practice with different rational functions will help cement this technique.
Integration Techniques
Integration is a fundamental concept in calculus, serving as the inverse operation to differentiation. There are various integration techniques employed to calculate the area under curves, among other applications. In the given problem, we use multiple integration techniques to solve the integral.

First, we have a straightforward polynomial term \(\int x dx\), which is integrated directly using the power rule. Next, there's the natural logarithm \(\ln|x|\), which comes from integrating the term \(\frac{1}{9x}\) that was a result of our partial fraction decomposition. Lastly, we use substitution to integrate the term \(\frac{-9x}{x^2+9}\), considering the derivative of \(x^2 + 9\) is \(2x\), which is present in the numerator.

When learning different techniques, students should:
  • Master basic integration formulas such as the power rule, concepts of the \(\ln\) function, trigonometric integrals, and substitution.
  • Recognize the appropriate technique for the term being integrated (e.g., substitution, partial fractions, by parts).
  • Understand that often, complex integrals may require a combination of methods for a successful solution.
Regular practice with various integrals enhances proficiency and ensures a deeper understanding of when and how to apply each technique effectively.
Polynomial Long Division
When the degree of the numerator is equal to or greater than the degree of the denominator in a rational function, polynomial long division must be employed before partial fraction decomposition can take place. This process is akin to long division with numbers and is used to simplify the rational function.

In our example, we initially faced the function \(\frac{x^4+1}{x^3+9x}\). We first considered polynomial long division to reduce the degree of the numerator. The quotient was \(x\) with a remainder of \(1 - 9x^2\), yielding a new integral of \(\int x + \frac{1-9x^2}{x^3+9x}dx\).

For better comprehension:
  • Get comfortable with the long division algorithm; familiarity with the process can make complex divisions more manageable.
  • Always check if long division is necessary by comparing the degrees of the numerator and the denominator.
  • Understand that the result of the division will be a polynomial plus a fraction with a lesser degree, ready for partial fraction decomposition if needed.
Fine-tuning skills in polynomial long division is beneficial for a strong foundation in calculus and simplifying more challenging integral problems.

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Most popular questions from this chapter

A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t),\) the Laplace transform is a new function \(F(s)\) defined by $$F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$ where we assume s is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated using integration by parts: $$F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1}$$ Verify the following Laplace transforms, where a is a real number. $$f(t)=\sin a t \rightarrow F(s)=\frac{a}{s^{2}+a^{2}}$$

Determine whether the following integrals converge or diverge. $$\int_{0}^{1} \frac{d x}{\sqrt{x^{1 / 3}+x}}$$

The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system. $$\int x \sin ^{-1} 2 x d x$$

Estimating error Refer to Theorem 8.1 in the following exercises. Let \(f(x)=\sqrt{\sin x}\) a. Find a Simpson's Rule approximation to \(\int_{1}^{2} \sqrt{\sin x} d x\) using \(n=20\) subintervals. b. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1. (Hint: Use the fact that \(\left.\left|f^{(4)}(x)\right| \leq 1 \text { on }[1,2] .\right)\)

Let \(a>0\) and \(b\) be real numbers. Use integration to confirm the following identities. (See Exercise 73 of Section 8.2) a. \(\int_{0}^{\infty} e^{-a x} \cos b x d x=\frac{a}{a^{2}+b^{2}}\) b. \(\int_{0}^{\infty} e^{-a x} \sin b x d x=\frac{b}{a^{2}+b^{2}}\)
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