**Question 3**If the substitution, $ u = \frac{x}{2} $ is made, the integral \[\int \left(\frac{1 - \frac{x}{2}}{x}\right)^2 \, dx \]is transformed into:- $\int \frac{(1-u)^2}{2u} \, du$- $\int \frac{(1-u)^2}{4u} \, du$- $2 \int \frac{(1-u)^2}{u} \, du$- $\int \frac{(1-u)^2}{u} \, du$---Click "Save and Submit" to save and submit. Click "Save All Answers" to save all answers.

Consider the integral ∫1−x22xdx

Answered: **Question 3** If the substitution, \(…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Expert Solution

Step 1

Consider the integral

$\int \frac{{(1 - \frac{x}{2})}^{2}}{x} d x$

Step 2

$\int \frac{{(1 - \frac{x}{2})}^{2}}{x} d x$

substitute

$\begin{array}{l} u = \frac{x}{2} \\ differentiate with respect to x \\ \frac{d u}{d x} = \frac{1}{2} \\ d u = \frac{d x}{2} \\ 2 d u = d x \end{array}$

Now the integral becomes

$\begin{array}{l} \int \frac{{(1 - \frac{x}{2})}^{2}}{x} d x = \int \frac{{(1 - u)}^{2}}{2 u} (2 d u) (∵ u = \frac{x}{2} \Rightarrow x = 2 u) \\ = \int \frac{{(1 - u)}^{2}}{2 u} (2 d u) \\ = \int \frac{{(1 - u)}^{2}}{u} d u \\ \int \frac{{(1 - \frac{x}{2})}^{2}}{x} d x = \int \frac{{(1 - u)}^{2}}{u} d u \end{array}$