/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 Why is the Substitution Rule ref... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 2

Why is the Substitution Rule referred to as a change of variables?

Short Answer

Expert verified
Answer: The solution to the integral is $\frac{1}{2}(x^3 + x^2 + 1)^2 + C$.

Step by step solution

01

Identify the given integral and choose a substitution

Consider the following integral as an example: \[ \int (3x^2 + 2x)(x^3 + x^2 + 1) dx \] In this case, we see that the function's derivative appears in the integral, which is a good indication that substitution could help simplify it. Let's choose the substitution: \[ u = x^3 + x^2 + 1 \]
02

Calculate the derivative of the chosen substitution

Now, we need to find the derivative of the chosen substitution with respect to x, which is: \[ \frac{du}{dx} = 3x^2 + 2x \]
03

Replace dx in terms of du

We have to express dx in terms of du to perform the change of variables completely. Rearrange the derivative equation to get dx and replace it in the integral: \[ dx = \frac{du}{(3x^2 + 2x)} \] Now, substitute u and dx in the original integral: \[ \int (3x^2 + 2x)(x^3 + x^2 + 1) dx = \int \frac{(3x^2 + 2x)}{(3x^2 + 2x)}u du \]
04

Simplify and solve the integral

The integral becomes much simpler after the substitution: \[ \int \frac{(3x^2 + 2x)}{(3x^2 + 2x)}u du = \int u du \] Now, it is easy to solve the integral: \[ \int u du = \frac{1}{2}u^2 + C \]
05

Replace the new variable with the original variable

In the final step, replace the new variable (u) with the original variable (x) using the substitution defined in Step 1: \[ \frac{1}{2}u^2 + C = \frac{1}{2}(x^3 + x^2 + 1)^2 + C \] This shows the process of the Substitution Rule or change of variables. By substituting one variable for another, we were able to simplify and solve the integral more easily. The integral solved using the Substitution Rule is: \[ \int (3x^2 + 2x)(x^3 + x^2 + 1) dx = \frac{1}{2}(x^3 + x^2 + 1)^2 + C \]

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

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

Suppose \(f\) is continuous on \([a, b]\) with \(f^{\prime \prime}(x)>0\) on the interval. It can be shown that $$(b-a) f\left(\frac{a+b}{2}\right) \leq \int_{a}^{b} f(x) d x \leq(b-a) \frac{f(a)+f(b)}{2}$$. a. Assuming \(f\) is nonnegative on \([a, b],\) draw a figure to illustrate the geometric meaning of these inequalities. Discuss your conclusions. b. Divide these inequalities by \((b-a)\) and interpret the resulting inequalities in terms of the average value of \(f\) on \([a, b]\).

Use a change of variables to evaluate the following integrals. $$\int \frac{e^{2 x}}{e^{2 x}+1} d x$$

Use a change of variables to evaluate the following integrals. $$\int\left(x^{3 / 2}+8\right)^{5} \sqrt{x} d x$$

Find the following integrals. $$\int \frac{x}{\sqrt[3]{x+4}} d x$$

Morphing parabolas The family of parabolas \(y=(1 / a)-x^{2} / a^{3}\) where \(a>0,\) has the property that for \(x \geq 0,\) the \(x\) -intercept is \((a, 0)\) and the \(y\) -intercept is \((0,1 / a) .\) Let \(A(a)\) be the area of the region in the first quadrant bounded by the parabola and the \(x\) -axis. Find \(A(a)\) and determine whether it is an increasing, decreasing, or constant function of \(a\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.