×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

18

0

3

# Class Note for MATH 250A with Professor Lega at UA

Marketplace > University of Arizona > Class Note for MATH 250A with Professor Lega at UA

No professor available

These notes were just uploaded, and will be ready to view shortly.

Either way, we'll remind you when they're ready :)

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

COURSE
PROF.
No professor available
TYPE
Class Notes
PAGES
3
WORDS
KARMA
25 ?

## Popular in Department

This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 18 views.

×

## Reviews for Class Note for MATH 250A with Professor Lega at UA

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 02/06/15
Calculus and Differential Equations l MATH 250 A Methods of integration ll Methods o tegration Calculus and Differential Equations The method of partial fractions 0 The purpose of the method of partial fractions is to find antiderivatives of rational functions ie functions of the form PX f X where P and Q are ol nomials QM p y o The method involves three steps 0 If d P 2 d Q first use long division and re write f as d H lt d Q7 where N and H are polynomials Then apply the method to the rational function HXQX 9 If d P lt d Q find the partial fraction decomposition of PXQX e Integrate each of the terms appearing in the partial fraction decomposition of f to obtain an antiderivative of f Methods of integration Calculus and Differential Equations The method of artial fractions continued a To do this we need to be able to perform each of the steps separately They are 0 Long division of polynomials 9 Partial fraction decomposition of PXQX where d P lt d Q 9 Integration of terms that typically appear in a decomposition into partial fractions Such terms are of the form A and BxC Xia 24rbltcquot7 where n 2 1 and X2 bx c is irreducible 0 Example for step 1 Divide X3 by X2 3X 2 tegration Calculus and Differential Equations Xles of aplication 7 i 0 We have already used partial fractions when solving the logistic equation 9 Solve the following differential equation y1y272y3 dX y25 39 9 Solve the differential equation dX y4 with the following initial conditions 0 y0 2 9 y0 1 Methods of integration Calculus and Differential Equations Trigonometric substitutions Trigonometric substitutions take advantage of known algebraic relationship between a Sines cosines and tangents cos20 l sin20 1 d cos0 7 sin0 d 2 tanW 7 1tan 0 7 d E sin0 7 cos0 1 cos20 o Hyperbolic sines cosines and tangents cosh20 7 sinh20 1 i cosh0 sinh0 i sinh0 cosh0 d0 d0 d 2 1 tanhW 7 17tanh 0 7 W Trigonometric substitutions continued 0 For integrands that involve 32 7 X2 3 gt 0 note that le g a and try the substitution X asin0 9 Since the integrand will involve cos20 and the dX will be given by dX acos0 d0 one can expect to be able to simplify the integral after such a substitution 0 Example Show that dX arcsin C dX m 0 Similarly for integrands that involve X2 7 32 a gt 0 one can change variables so that X gt 0 and then try X acosh0 since X2 2 32 9 Examples Show that X2 7 32 dX can be written as a2 sinh20 d0 after a substitution Methods of Integration Calculus and Differential Equations Calculus and Differential Equations Methods of integration Halfangle substitutions o Half angle substitutions are useful to find antiderivatives of products andor ratios of sines and cosines 0 Indeed let ttan02 Then 1722 1t27 2t cos0 1 t2 sin0 7 dt 51 d0 0 A product or ratio of sines and cosines will thus be transformed into a rational function of t which we know how to integrate using partial fractions 0 t C 3 2gtl Calculus and Differential Equations 0 Example Show that ln i sin0 Methods of integration Partial fraction decomposition P X X into partial fractions proceed as follows To decompose the rational function where d P lt d Q 0 Factor the denominator QX into terms of the form X 7 a and X2 bxc where n 21 and X2 bxc is irreducible 9 For each factor of the form X7 a the partial fraction decomposition of PXQX will include terms of the form A1 A2 X7a7 X7a27 A1 7 Xiay39v39 An 7 Xia 39 9 To find An multiply by X 7 a and set X a into the resulting equation 0 To find the Aj s j y n multiply by X7 a and substitute in appropriate values of X Methods of integration Calculus and Differential Equations Partial fraction decomposition continued 6 For each factor of the form X2 bX c th e partial fraction decomposition of PXQX will include terms of the form 81X C1 BjXCj Xzbxc7 7 X2 l bxcj7 H an C 7 X2bxc 39 9 To find the Bj s and Cj s multiply by X2 I bx c expand and equate the coefficients of the various powers of X in both sides of the resulting equation Example Find the partial fraction decomposition of X25 XI x1X272x339 Mamas o tegration Calculus and Differential Equations Integration of a partial fraction decomposition Typical terms in a partial fraction decomposition are of the form A and BxC Xia X2 bxc 39 0 Terms of the form n gt1 X i a 9 I7 1 then X a dx lnlxial C o lfngt 1 then Methods of integration Calculus and Differential Equations Integration of a partial fraction decomposit B C 1 n 2 1 X2 bX c 0 Compare the numerator to the derivative of X2 9 Terms of the form ion continued bxc BxC dx 2xbiCdX X2bxc X2bxc 5 D dx T 2 U L lt2bltc 7 bB whereuX2bxcand DCiT 9 Thus we can integrate provided we know how to find an antiderivative of lX2 bX c 9 Note that since X2 bx c is irreducible one can write X2bXCltX tegration b2 d2 where a 2 c 7 4 Calculus and Differential Equations Integration of a partial fraction decomposition continued 1 b n etu Then 9 To integrate x 2d2 d M dX i 1 du lltx 92 NZ T d2 2 1 0 lfn1then dX fl du fl 1 i C Xg2d2Tdu21Tdarcan d 2d du 1u2 u2Llln W c052 20 d0 Alternatively integrate by parts and find a recursive formula o If n gt 1 let 9 arctanu Then d0 and Methods of integration Calculus and Differential Equations

×

×

### BOOM! Enjoy Your Free Notes!

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Bentley McCaw University of Florida

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Allison Fischer University of Alabama

#### "I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over \$600 per month. I LOVE StudySoup!"

Steve Martinelli UC Los Angeles

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Parker Thompson 500 Startups

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com