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Chapter 8: Problem 48

Evaluate each integral in Exercises \(47-52\) by reducing the improper fraction and using a substitution (if necessary) to reduce it to standard form. $$ \int \frac{x^{2}}{x^{2}+1} d x $$

Short Answer

Expert verified
The integral evaluates to \( x - \tan^{-1}(x) + C \).

Step by step solution

01

Identify the Type of Integral

The given integral is \( \int \frac{x^{2}}{x^{2}+1} \, dx \). We recognize this as an improper fraction since the degree of the numerator is the same as the degree of the denominator.
02

Perform Polynomial Long Division

Since the numerator and denominator are of the same degree, divide the numerator \( x^2 \) by the denominator \( x^2 + 1 \). This results in:\[x^2 = (1)(x^2 + 1) - 1\]So, we rewrite:\[\frac{x^2}{x^2 + 1} = 1 - \frac{1}{x^2 + 1}\]
03

Separate the Integral

Substitute the result from the division into the integral:\[\int \left( 1 - \frac{1}{x^2+1} \right) \, dx = \int 1 \, dx - \int \frac{1}{x^2+1} \, dx\]This allows us to split the integral into two parts.
04

Integrate Each Part Separately

Integrate \( \int 1 \, dx \) which equals \( x \), and integrate \( \int \frac{1}{x^2+1} \, dx \), which equals \( \tan^{-1}(x) \) because the integral of \( \frac{1}{a^2 + x^2} \) is \( \frac{1}{a} \tan^{-1}\left( \frac{x}{a} \right) \) resulting in:\[\int 1 \, dx = x\]and\[\int \frac{1}{x^2+1} \, dx = \tan^{-1}(x)\]
05

Combine the Results

Combine the results of the integrals:\[x - \tan^{-1}(x) + C\]where \( C \) 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.

Improper Fraction
An improper fraction in mathematics is one where the degree (or highest power) of the numerator is greater than or equal to the degree of the denominator. This case is noticeably different from a proper fraction, where the numerator's degree is less than the denominator's. In the example given, the integral \( \int \frac{x^2}{x^2+1} \, dx \) features an improper fraction because both the numerator and the denominator have a degree of 2.
  • Numerator: \( x^2 \) (degree 2)
  • Denominator: \( x^2 + 1 \) (degree 2)
Understanding this is crucial because improper fractions often require simplification techniques like polynomial long division to be integrable more easily.
Polynomial Long Division
Polynomial long division is a method similar to numerical long division used to divide polynomials. When dealing with improper fractions in integrals, polynomial long division can help simplify the expression by separating it into more manageable pieces.
In our example, dividing \( x^2 \) by \( x^2 + 1 \) simplifies the process. Keeping track of what you're doing is easier if you remember:
  • Divide the leading term of the numerator by the leading term of the denominator.
  • Multiply the entire divisor by the result from the first division and subtract the result from the original numerator.
  • Repeat the steps if necessary.
After completing the division, the integral \( \int \frac{x^2}{x^2+1} \, dx \) can be rewritten as \( \int \left( 1 - \frac{1}{x^2 + 1} \right) \, dx \), making integration straightforward.
Integration by Substitution
Integration by substitution is a valuable tool when simplifying integrals, particularly when an integral includes a function and its derivative. This method changes the variable of integration to transform the integral into a simpler form. However, in this particular exercise, substitution is not needed for the polynomial division part, but it is often used elsewhere in calculus.
The key steps for integration by substitution include:
  • Select a substitution \( u = g(x) \) for simplifying the integral.
  • Determine \( du \) by differentiating \( g(x) \).
  • Rewrite the entire integral in terms of \( u \).
  • Integrate and then substitute back the original variable.
Once you grasp substitution, it becomes a powerful technique for handling complex integrals.
Arctangent Function
The arctangent function, denoted as \( \tan^{-1}(x) \), is an inverse trigonometric function that limits angles within \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This function frequently appears in integrals involving \( \frac{1}{x^2+a^2} \), as it's the antiderivative of such expressions.
In our step-by-step solution, after simplifying the integral to \( \int \left( 1 - \frac{1}{x^2+1} \right) \, dx \), integrating the second part \( \int \frac{1}{x^2+1} \, dx \) leads to \( \tan^{-1}(x) \). This is because:
  • The integral of \( \frac{1}{a^2 + x^2} \) is \( \frac{1}{a} \tan^{-1}\left( \frac{x}{a} \right) \), our specific case where \( a = 1 \).
  • This results in the function \( \tan^{-1}(x) \), seamlessly handled in calculus problems.
Mastering the arctangent function helps ease many integral evaluations, particularly those with expressions reminiscent of the standard mentioned formula.

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Most popular questions from this chapter

In Exercises \(15-26,\) estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than \(10^{-4}\) by ( a ) the Trapezoidal Rule and (b) Simpson's Rule. (The integrals in Exercises \(15-22\) are the integrals from Exercises \(1-8 .\) ) $$ \int_{1}^{3}(2 x-1) d x $$

Use a CAS to perform the integrations. a. Use a CAS to evaluate $$\int_{0}^{\pi / 2} \frac{\sin ^{n} x}{\sin ^{n} x+\cos ^{n} x} d x$$ where \(n\) is an arbitrary positive integer. Does your CAS find the result? b. In succession, find the integral when \(n=1,2,3,5,7 .\) Comment on the complexity of the results. c. Now substitute \(x=(\pi / 2)-u\) and add the new and old integrals. What is the value of $$\int_{0}^{\pi / 2} \frac{\sin ^{n} x}{\sin ^{n} x+\cos ^{n} x} d x ?$$ This exercise illustrates how a little mathematical ingenuity solves a problem not immediately amenable to solution by a CAS.

Use a CAS to perform the integrations. Evaluate the integrals a. \(\int x \ln x d x \) b. \(\int x^{2} \ln x d x\) c. \(\int x^{3} \ln x d x\) d. What pattern do you see? Predict the formula for \(\int x^{4} \ln x d x\) and then see if you are correct by evaluating it with a CAS. e. What is the formula for \(\int x^{n} \ln x d x, n \geq 1 ?\) Check your answer using a CAS.

Solve the initial value problems in Exercises \(37-40\) for \(y\) as a function of \(x .\) $$ x \frac{d y}{d x}=\sqrt{x^{2}-4}, \quad x \geq 2, \quad y(2)=0 $$

Show that if \(f(x)\) is integrable on every interval of real numbers and \(a\) and \(b\) are real numbers with \(a
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