Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

u2du

Short Answer

Expert verified

The three integrals sinx(cosx)2dx,(lnx)2xdxand(2x+1)(x2+x)dxwill have form u2duafter a substitution of variables.

Step by step solution

01

Step 1. Given Information 

For given integral we have to write down three integrals that will have that form after a substitution of variables.
u2du

02

Step 2. The first integrals that will have that form after a substitution of variables.

Let

u=cosxdudx=sinxdu=sinxdx

This substitution changes the integral into

sinx(cosx)2dx

03

Step 3. The first integrals that will have that form after a substitution of variables.

Let

u=lnxdudx=1xdu=1xdx

This substitution changes the integral into

(lnx)2xdx

04

Step 4. The first integrals that will have that form after a substitution of variables.

Let

u=x2+xdudx=2x+1du=(2x+1)dx

This substitution changes the integral into

(2x+1)(x2+x)dx

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 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

For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

eudu

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) An integral with which we could reasonably apply trigonometric substitution with x=tanu.

(b) An integral with which we could reasonably apply trigonometric substitution with x=4secu.

(c) An integral with which we could reasonably apply trigonometric substitution with x2=3sinu.

Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial x2-x+3 as the quotient above the top line, and the polynomial 3x − 1 at the bottom as the remainder. Thenp(x)=___andp(x)q(x)=____

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingx2+45/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2a2.

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

xx2-1dx

See all solutions

Recommended explanations on Math Textbooks

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