Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 12: Problem 1048

If \(a<0

Short Answer

Expert verified
The short answer is: \(b\int_a^b\frac{|x|}{x} dx = ab + b^2\)

Step by step solution

01

Rewrite the function with the absolute value function

The given function is: \(|x|/x\) We can rewrite the function without using the absolute value by considering the cases when x is positive or negative. If x < 0, then |x| = -x, and |x|/x = -1 If x > 0, then |x| = x, and |x|/x = 1 Now, we rewrite the function as: \[f(x) = \begin{cases} -1 & \text{if } x < 0 \\ 1 & \text{if } x \geq 0 \end{cases} \]
02

Break the integral using the cases

Now we will break the integral into two parts using the function definition above and the given bounds (a, b). Since a < 0 and b > 0, the integral can be split at x = 0: \[b\int_a^b\frac{|x|}{x} dx = b\int_a^0\frac{-x}{x} dx + b\int_0^b\frac{x}{x} dx\]
03

Evaluate the integrals

Now, we evaluate the integrals for each part: \[b\int_a^0\frac{-x}{x} dx = b\int_a^0(-1) dx = -b[x]_a^0 = -b(0-a) = ab\] \[b\int_0^b\frac{x}{x} dx = b\int_0^b 1 dx = b[x]_0^b = b(b-0) = b^2\]
04

Combine the results and simplify

Now, we combine the results from step 3 and simplify: \(b\int_a^b\frac{|x|}{x} dx = ab + b^2\)
05

Identify the correct option

Now, we check the given multiple-choice options to find the correct answer: (a) a + b: This cannot be the correct answer since our result contains an extra b term. (b) b - a: This also cannot be the correct answer since our result contains a squared term. (c) a - b: This cannot be the correct answer since our result contains a squared term. (d) \((b - a) / 2\): This also cannot be the correct answer since our result does not divide the expression by 2. None of the options match the result we obtained (ab + b^2). There is a possibility of an error in the given options, so we conclude with our obtained result: \[b\int_a^b\frac{|x|}{x} dx = ab + b^2\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The area bounded by the curves \(y=x^{2}\) and \(y=|x|\) is...... (a) 1 Sq. unit (b) 2 Sq. unit (c) \((1 / 3)\) Sq. unit (d) \((2 / 3)\) Sq. unit

The value of the integral \(\pi \int_{-\pi}\left[\left(\cos ^{2} x\right) /\left(1+3^{x}\right)\right] d x\) is \(\ldots \ldots\) (a) 0 (b) \((\pi / 4)\) (c) \((\pi / 2)\) (d) \(\pi\)

The value of the integral \({ }^{1} \int_{0} 2^{2 \mathrm{x}} \cdot 3^{-\mathrm{x}} \mathrm{dx}\) is \(\ldots \ldots \ldots\) (a) \(\log _{\mathrm{e}}(64 / 27)\) (b) \(\log _{\mathrm{e}}(27 / 64)\) (c) \(\log _{(3 / 4)}\) e (d) \(\log _{(64 / 27)} \mathrm{e}\)

\((\pi / 2) \int_{(\pi / 4)} \sqrt{(1-\sin 2 x) d x}\) is equal to \(\ldots \ldots \ldots\) (a) \(\sqrt{2}+1\) (b) \(\sqrt{2}-1\) (c) \(1-\sqrt{2}\) (d) 0

The value of \((e)^{2} \int_{e}[d x /(\log x)]-2 \int_{1}\left(e^{x} / x\right) d x\) is...... (a) \(\mathrm{e}^{2}\) (b) e (c) \((1 / e)\) (d) 0
See all solutions

Recommended explanations on English Textbooks

History of English Language

Read Explanation

Graphology

Read Explanation

Essay Prompts

Read Explanation

Creative Story

Read Explanation

Research and Composition

Read Explanation

Blog

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free