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 x+2x2+4x32dxthe following two ways:

(a) with the substitution u=x2+4x;

(b) by completing the square and then applying the trigonometric substitution x + 2 = 2 sec u.

Short Answer

Expert verified

Part (a) The solution of the integral is -1x2-4x+C.

Part (b) The solution of the integral is -1x2-4x+C.

Step by step solution

01

Part (a) Step 1. Given Information.

The given integral isx+2x2+4x32dx.

02

Part (a) Step 2. Solve. 

We have to solve the integral with the substitution u=x2+4x,so the derivative isdu=2x+4dx.

Let's solve the integral by substituting u,

x+2x2+4x32dx=x+2u32du2x+4=12duu32=12-21u+C=-1u+CSubstitutebacku,=-1x2+4x+C

03

Part (b) Step 1. Solve.

We have to solve it first by completing the square.

So, let's complete the square,

x2+4x=x2+4x+22-22=x+22-4

04

Part (b) Step 2. Solve.

Now, we have to apply the trigonometric substitution x+2=2secu.

So, Let's solve the integral by substituting,

x+2x2+4x32dx=x+2x+22-432dx=2secu2secu2-4322secutanudu=4sec2utanu4sec2u-432du=4sec2utanu8sec2u-132du=12sec2utanusec2u-132duUsetheidentity1+tan2x=sec2x,=12sec2utanutan2u32du=12sec2utan2udu=12csc2udu=12-cotu+CSubstitutebacku,=-12cotsec-1x+22+C=-1x2-4x+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

Most popular questions from this chapter

True/False: Determinewhethereachofthestatementsthat follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: f(x)=x+1x-1is a proper rational function.

(b) True or False: Every improper rational function can be expressed as the sum of a polynomial and a proper rational function.

(c) True or False: After polynomial long division of p(x) by q(x), the remainder r(x) has a degree strictly less than the degree of q(x).

(d) True or False: Polynomial long division can be used to divide two polynomials of the same degree.

(e) True or False: If a rational function is improper, then polynomial long division must be applied before using the method of partial fractions.

(f) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Ax2+Bx-3

(g) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Bx+Cx2+Ax-3.

(h) True or False: Every quadratic function can be written in the formA(x-k)2+C

Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution withx=asinuor x=atanu? Why do we need to think about the unit circle after trigonometric substitution with x=asecu?

Solve given definite integral.

04x3x2+4dx

Solve the integral:ln3xdx

Show by differentiating (and then using algebra) that cotsin1xand 1x2xare both antiderivatives of 1x21x2. How can these two very different-looking functions be an antiderivative of the same function?

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