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

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

xln(ex2+1)dx

Short Answer

Expert verified

The solution of the given integral is xln(ex2+1)dx=12eln(ex2+1)+C.

Step by step solution

01

Step 1. Given Information 

Solving the given integrals.

xln(ex2+1)dx

02

Step 2. Solving the given integral using substitution method. 

Let

u=ln(ex2+1)dudx=2xex2+1du=2xex2+1dxex2+12du=xdx

03

Step 3. In this substitution after differentiation not at all in the form of f'(u(x))u'(x).

A clever change of variables will allow us to rewrite the integral so that it can be algebraically simplified.

u=ln(ex2+1)eu=ex2+1

eu2du=xdx

04

Step 4. This substitution changes the integral into 

xln(ex2+1)dx=12euduxln(ex2+1)dx=12eu+Cxln(ex2+1)dx=12eln(ex2+1)+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 Vaia!

One App. One Place for Learning.

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

Get started for free

Study anywhere. Anytime. Across all devices.

Sign-up for free