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Chapter 7: Problem 48

(a) Make an appropriate \(u\) -substitution, and then use the Endpaper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution), and then confirm that the result is equivalent to that in part (a). $$ \int x \ln \left(2+x^{2}\right) d x $$

Short Answer

Expert verified
The integral is \( \frac{1}{2} \left( (2+x^2) \ln(2+x^2) - (2+x^2) \right) + C \).

Step by step solution

01

Choose the substitution

Identify a suitable function for substitution. Set \( u = 2 + x^2 \), which simplifies the integral.
02

Differentiate u

Compute the derivative of \( u \) with respect to \( x \). This gives \( \frac{du}{dx} = 2x \). Solve for \( dx \) to substitute later: \( dx = \frac{du}{2x} \).
03

Substitute into the integral

Replace variables in the integral using the substitutions: \( \int x \ln(u) \cdot \frac{du}{2x} \). This simplifies to \( \frac{1}{2} \int \ln(u) \, du \).
04

Integrate with respect to u

Use the Endpaper Integral Table to find \( \int \ln(u) \, du = u \ln(u) - u + C \). Thus, our integral becomes \( \frac{1}{2} \left( u \ln(u) - u \right) + C \).
05

Substitute back for x

Replace \( u \) with the expression \( 2 + x^2 \) to revert to the original variable: \( \frac{1}{2} \left( (2+x^2) \ln(2+x^2) - (2+x^2) \right) + C \).
06

Verify with CAS

Use a Computer Algebra System (CAS) to evaluate the integral directly: \( \int x \ln(2 + x^2) \, dx = \frac{1}{2} \left( (2+x^2) \ln(2+x^2) - (2+x^2) \right) + C \).
07

Confirm equivalence

Ensure that the result from the CAS matches the expression obtained through substitution, confirming the solution is consistent.

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Key Concepts

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

u-substitution
The "u-substitution" method is a powerful technique in integral calculus to simplify integrals for easier evaluation. It involves changing the variable of integration to transform a complex integral into a simpler form.
This technique is often useful when dealing with composite functions, such as functions of functions.Here's how it works:
  • Identify a substitution: Choose a part of the integrand to replace with a new variable, typically denoted as \( u \).
  • Differentiate the substitution: Differentiate \( u \) with respect to the variable of integration (usually \( x \)) to find \( \frac{du}{dx} \).
  • Rewrite the integral: Replace all occurrences of the expressions in the original integral with the new variable \( u \) and \( du \).
  • Integrate: Solve the new, usually simpler, integral with respect to \( u \).
  • Back-substitute: Return to the original variable by replacing \( u \) with the corresponding expression in terms of \( x \).
In the given exercise, the substitution \( u = 2 + x^2 \) allows us to simplify the integral involving logarithms, making the integration process more manageable.
integral calculus
Integral calculus is the study of integrals and their properties. The integral of a function represents the accumulation of quantities and can be seen as the area under a curve.There are two main types of integrals:
  • Definite integrals: These have limits of integration and yield a numeric result representing the net area under the curve.
  • Indefinite integrals: These do not have limits of integration and represent a family of functions, including a constant \( C \), since integration is the reverse process of differentiation.
Evaluating integrals analytically often requires special techniques like u-substitution, integration by parts, or trigonometric identities. In this specific case, the use of u-substitution simplified the complex logarithmic integral to a form that can be integrated using a formula from the Endpaper Integral Table. Remember, integral calculus not only allows us to find areas but also has applications in physics, engineering, and beyond.
computer algebra system (CAS)
A Computer Algebra System (CAS) is a software tool that facilitates symbolic mathematics, allowing users to perform complex algebraic tasks with ease.CAS can be a powerful aid in verifying solutions to intricate calculus problems:
  • Automated calculations: These systems can handle complicated manipulations swiftly and accurately, even those that might be prone to human error.
  • Symbolic manipulations: Unlike numerical calculators, CAS can solve equations symbolically, displaying exact expressions instead of decimal approximations.
  • Verification tool: In educational settings, it serves as a "second pair of eyes" ensuring that manual computations are correct, as demonstrated in the given exercise.
In this exercise, CAS was employed to verify the solution obtained from u-substitution. By directly evaluating the integral \( \int x \ln(2+x^2) \ dx \) using CAS and obtaining the same result, the consistency and correctness of the u-substitution method were confirmed, showcasing CAS's reliability and functionality in solving integrals.

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Most popular questions from this chapter

(a) Make an appropriate \(u\) -substitution of the form \(u=x^{1 / n}\) or \(u=(x+a)^{1 / n}\), and then evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$ \int x \sqrt{x-2} d x $$

Use any method to find the area of the surface generated by revolving the curve about the \(x\) -axis. $$ y=1 / x, 1 \leq x \leq 4 $$

Use any method to find the volume of the solid generated when the region enclosed by the curves is revolved about the \(y\) -axis. $$ y=e^{-x}, \quad y=0, x=0, x=3 $$

(a) Make an appropriate \(u\) -substitution of the form \(u=x^{1 / n}\) or \(u=(x+a)^{1 / n}\), and then evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$ \int \frac{1+\sqrt{x}}{1-\sqrt{x}} d x $$

Use any method to solve for \(x\). $$ \int_{2}^{x} \frac{1}{t(4-t)} d t=0.5,2
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