/*! 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 39 Use a change of variables to eva... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 39

Use a change of variables to evaluate the following definite integrals. $$\int_{0}^{1} 2 x\left(4-x^{2}\right) d x$$

Short Answer

Expert verified
Question: Evaluate the given definite integral using a change of variables: $$\int_{0}^{1} 2 x\left(4-x^{2}\right) d x$$ Answer: After applying the change of variables and evaluating the integral, we find that the value of the definite integral is $$-\frac{7}{2}$$.

Step by step solution

01

Find an appropriate substitution

Examining the expression inside the integral, we can choose the substitution \(u = 4 - x^2\), as it simplifies the expression. Differentiating with respect to x, we get: $$\frac{d u}{d x} = -2x$$ Now, we solve for \(dx\): $$d x = \frac{-d u}{2 x}$$
02

Change the limits of integration

We need to find the new limits of integration in terms of the variable u: When \(x=0\), \(u=4-0^2=4\) When \(x=1\), \(u=4-1^2=3\) So our new limits of integration are from 3 to 4.
03

Rewrite the integral in terms of u

Now let's rewrite the integral using the substitution and the new limits of integration: $$\int_{3}^{4} 2x \left(4-x^2\right) \frac{-du}{2 x}=-\int_{3}^{4} u d u$$
04

Evaluate the definite integral

Now, we can evaluate the integral as follows: $$-\int_{3}^{4} u d u = -\frac{1}{2}u^2\Big|_3^4 = -\frac{1}{2}((4)^2-(3)^2)$$ $$=-\frac{1}{2}(16-9) = -\frac{1}{2}(7) = -\frac{7}{2}$$ So, the value of the original definite integral $$\int_{0}^{1} 2 x\left(4-x^{2}\right) d x$$ is \(-\frac{7}{2}\).

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.

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

Multiple substitutions Use two or more substitutions to find the following integrals. $$\int x \sin ^{4}\left(x^{2}\right) \cos \left(x^{2}\right) d x$$ (Hint: Begin with \(u=x^{2}\), then use \(v=\sin u .)\)

General results Evaluate the following integrals in which the function \(f\) is unspecified. Note \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f\). Assume \(f\) and its derivatives are continuous for all real numbers. $$\int 2\left(f^{2}(x)+2 f(x)\right) f(x) f^{\prime}(x) d x$$

Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\) and interpret the results using the graphs of part \((b)\) $$f(x)=\frac{1}{x} ; a=1, b=4, c=6$$

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

Integral of \(\sin ^{2} x \cos ^{2} x\) Consider the integral \(I=\int \sin ^{2} x \cos ^{2} x d x\) a. Find \(I\) using the identity \(\sin 2 x=2 \sin x \cos x\) b. Find \(I\) using the identity \(\cos ^{2} x=1-\sin ^{2} x\) c. Confirm that the results in parts (a) and (b) are consistent and compare the work involved in each method.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

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.