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Chapter 8: Problem 24

In Exercises \(21-32,\) express the integrand as a sum of partial fractions and evaluate the integrals. $$\int \frac{8 x^{2}+8 x+2}{\left(4 x^{2}+1\right)^{2}} d x$$

Short Answer

Expert verified
Decompose into partial fractions and integrate each term separately.

Step by step solution

01

Express the Integrand as a Partial Fraction

Begin by recognizing the structure of the denominator, which is \((4x^2 + 1)^2\). Since it is a repeated irreducible quadratic, the partial fraction decomposition involves terms of the form \(\frac{Ax + B}{4x^2 + 1}\) and \(\frac{Cx + D}{(4x^2 + 1)^2}\). Thus, set up:\[\frac{8x^2 + 8x + 2}{(4x^2 + 1)^2} = \frac{Ax + B}{4x^2 + 1} + \frac{Cx + D}{(4x^2 + 1)^2}\] and solve for \(A, B, C,\) and \(D\).
02

Solve for Coefficients

Multiply both sides by \((4x^2 + 1)^2\) to get:\[8x^2 + 8x + 2 = (Ax + B)(4x^2 + 1) + Cx + D\]Expand and collect terms, equating the coefficients on both sides:\[4Ax^3 + (4B + A)x^2 + (B)x + (B) = 8x^2 + 8x + 2\]Equate coefficients for respective powers of \(x\). This provides a system of equations to solve for \(A, B, C,\) and \(D\).
03

Integrate Each Partial Fraction Term

Once the coefficients \(A, B, C,\) and \(D\) are determined, rewrite the integrand using partial fractions, then separately integrate each term:\[\int \frac{Ax + B}{4x^2 + 1} \, dx + \int \frac{Cx + D}{(4x^2 + 1)^2} \, dx\]For \(\int \frac{Ax + B}{4x^2 + 1} \, dx\), use substitution techniques, and for \(\int \frac{Cx + D}{(4x^2 + 1)^2} \, dx\), consider using trigonometric substitution or other applicable techniques.
04

Compute the Integrals

For the term \(\int \frac{Ax + B}{4x^2 + 1} \, dx\), use substitution or recognize it as a standard arctangent form. For \(\frac{Cx + D}{(4x^2 + 1)^2}\), simplify using either substitution or integration by parts if applicable, taking into account simplifications from previous steps.
05

Combine the Results

The final solution requires summing up the integrals computed in the previous steps to get the complete integral of the original function:\[= \int \frac{8x^2 + 8x + 2}{(4x^2 + 1)^2} \, dx\] possibly plus a constant of integration if indefinite.

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Key Concepts

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

Integral Calculus
Integral calculus is all about finding the area under a curve or, more formally, the antiderivative of a function. It is a crucial part of mathematics that extends beyond polynomials and into various types of integrands, including rational functions. In this exercise, integral calculus is applied to solve an integral involving partial fractions.
Partial fraction decomposition is particularly useful in integral calculus when dealing with rational functions, which are ratios of polynomials.
  • Integral calculus helps simplify complex functions to integrable forms.
  • It transforms intricate rational functions into a series of simpler functions that can be easily integrated.
  • Understanding integral calculus techniques like substitution and integration by parts is essential for solving integrals.
This foundational aspect of calculus not only aids in computations but also in understanding the behavior and properties of functions over intervals.
Integration Techniques
When working with integrals, especially those that involve partial fractions, various integration techniques come into play. This exercise demonstrates the application of these techniques to solve the integral of a complex rational function.
One common technique is substitution, useful when a function within the integrand resembles the derivative of another function.
For example, in this exercise:
  • Substitution: It is often used when integrating terms like \(\frac{Ax + B}{4x^2 + 1}\), converting them to forms integral calculus can easily handle.
  • Trigonometric Substitution: Particularly effective for integrands involving quadratic forms like \(4x^2 + 1\), where substitution simulates trigonometric identities to simplify the integration process.
Mastering these techniques is pivotal because they allow you to decompose an intricate integral into manageable pieces, leading to solutions that are both elegant and practical.
Polynomial Division
Polynomial division is a precursor step in the process of partial fraction decomposition for integrals involving rational functions. When you have a numerator with a higher degree than the denominator, polynomial division becomes indispensable.
In the current exercise, the goal was to express the original integrand as a sum of simpler fractions:
  • If the numerator's degree is equal to or greater than the denominator's degree, polynomial division helps reduce it.
  • Through division, complicated expressions simplify to a sum of fractions where the numerator's degree is lesser than the denominator's.
  • Further simplification allows us to set up partial fractions easily, facilitating more straightforward integration.
By embracing polynomial division, students can transform unwieldy rational functions into expressions that are easily dissected and analyzed during integration.

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Most popular questions from this chapter

The error function The error function, $$ \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t $$ important in probability and in the theories of heat flow and signal transmission, must be evaluated numerically because there is no elementary expression for the antiderivative of \(e^{-t^{2}}\) . a. Use Simpson's Rule with \(n=10\) to estimate erf \((1) .\) b. In \([0,1]\) , $$ \left|\frac{d^{4}}{d t^{4}}\left(e^{-t^{2}}\right)\right| \leq 12 $$ Give an upper bound for the magnitude of the error of the estimate in part (a).

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{1}^{\infty} \frac{\sqrt{x+1}}{x^{2}} d x$$

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{4}^{\infty} \frac{d x}{\sqrt{x}-1}$$

About the infinite region in the first quadrant between the curve \(y=e^{-x}\) and the \(x\) -axis. Find the centroid of the region.

Suppose you toss a fair coin \(n\) times and record the number of heads that land. Assume that \(n\) is large and approximate the discrete random variable \(X\) with a continuous random variable that is normally distributed with \(\mu=n / 2\) and \(\sigma=\sqrt{n} / 2 .\) If \(n=400\) find the given probabilities. $$ \begin{array}{ll}{\text { a. } P(190 \leq X<210)} & {\text { b. } P(X<170)} \\\ {\text { c. } P(X>220)} & {\text { d. } P(X=300)}\end{array} $$
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