### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Class Note for MATH 250A at UA

### View Full Document

## 25

## 0

## Popular in Course

## Popular in Department

This 102 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 25 views.

## Reviews for Class Note for MATH 250A at UA

### What is Karma?

#### Karma is the currency of StudySoup.

#### 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

MATH 250a Fall Semester 2007 Section 2 J M Cushing Thursday September 27 httprnathariz0naeducushing250ahtrnl Chapter 7 Integration Sections 1 4 gt Calculation of antiderivatives integrals 1 Learn some fundamental properties of integrals 2 Use known differentiation formulas backwards 3 Use differentiation rules backwards chain rule substitution product rule integration by parts 4 Use algebraic manipulations eg partial fraction decomposition 5 Use tables of integrals 4 Use algebraic manipulations Partial Fraction Decomposition Section 74 in text 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions l fxadx 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 f dx X Cl wzx m 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions l fxadx wzx a d wlgtdwdx dx 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions l l fxadxfdw wzx a d wlgtdwdx dx 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions f 1 dxfdw1nwc X Cl wzx a d wlgtdwdx dx 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions f 1 dxlnx ac x a 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions f 1 dxlnx a l c x a x a for an integer n gt 1 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions f 1 dxlnx a l c x a x a wzx a lgtdwdx dx 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions fxiadlenlx al l c dw wzx a d wlgtdwdx dx 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions l l l l 1 fxadx lnx alc fWW fwndW lnw 6 wzx a d wlgtdwdx dx 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 1 1 lin fxadxlnx ac fmw1nx a c 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions l l 1 1 dx fxadx lnx al l c fltxagtn 1 ltxagtn1c 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions f 1 dxlnx a l c 1 c x a x aquot 1 nx an1 l fx2a2dx 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions l l l l dx fxadx lnx al l c fltxagtn 1nltxagtn1c l fx2a2dx 1 Recall f 2 dx arctan x c x 1 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions l l l l dx fxadx lnx al l c fltxagtn 1nltxagtn1c fxziazdxf 2 x12 a 2l a clx 1 Recall f 2 dx arctan x c x 1 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 1 1 1 01x fxadx lnx al l c fltxagtn 1nltxagtn1c l l l fx2a2dxfx2 dx 1 a 1 Recall dx arctan x c fx2l 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 1 1 1 01x fxadx lnx al l c fltxagtn 1nltxagtn1c fo l azdxz f 1 dx 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 1 1 1 01x fxadx lnx al l c fltxagtn 1nltxagtn1c 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 1 1 1 01x fxadx lnx al l c fltxagtn 1nltxagtn1c l l l fo l azdngfuZ l ldu 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 1 1 1 01x fxadx lnx al l c fltxagtn 1nltxagtn1c l l l l dxz duz arctanu l c fxz l a2 afu2l a 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 1 1 1 dx fxadx lnx al l c fltxagtn 1nltxagtn1c 1 f 2 2alxiarctan c xa a a 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions f 1 dxlnx a l c fdx 1 1 x a x a 1 1 lx an1 6 xa a a 1 f 2 2alxiarctan c fx2a2dx 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions f 1 dxlnx a l c fdx 1 1 x a x a 1 1 lx an1 6 xa a a 1 f 2 2alxiarctan c fx2a2dx 2 2 wx a 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions f 1 dxlnx a l c fdx 1 1 x a x a 1 1 lx an1 6 1 f 2 2alxiarctan c X 61 61 a x dx fx2a2 2 2 1 w x a 2xgtxdx dw E 2 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 1 1 1 dx fxadx lnx al l c fltxagtn 1nltxagtn1c 1 f 2 2alxiarctan c xa a a fx2a2dxfdw wx2a2 2xxdxldw 2 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 1 1 1 01x fxadx lnx al l c fltxagtn 1nltxagtn1c 1 f 2 2alxiarctan c xa a a x 1 1 1 f2 201x5f21w 1nwc 2xixdxldw 2 2 dx 2 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 1 1 1 dx fxadx lnx al l c fltxagtn 1nltxagtn1c f 21 2alxiarctan c f x alxz lnle i aZl i c xa a a 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions l l l delDIx aIlC Note denominator has single factor of x a has a simple root x a 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions f 1 dxlnx a l c fdx 1 1 x a x a 1 1 lx an1 6 f 21 2alxiarctan c f x alxz lnle i aZl i c x a a a xz l a2 Note denominator has n factors of x a has a multiple root x a 4 Use algebraic manipulations Partial Fraction Decomposition Useful for integrating rational functions First we look at some simpler rational functions 1 1 1 1 dx fxadx lnx al l c fltxagtn 1nltxagtn1c l l x x l dx arctan c dxz ln xz l a2 6 fx2a2 a a fx2a2 2 I I Note denominator has no real roots is not factorable 4 Use algebraic manipulations Partial Fraction Decomposition f2 23 dx x 1x1 4 Use algebraic manipulations Partial Fraction Decomposition 2 3 1 1 fx1 x21dx2fx1dx 3fx21dx 4 Use algebraic manipulations Partial Fraction Decomposition 2 3 1 1 fx1 x21dx2fx1dx 3fx21dx 21nx 1 3arctanxc 4 Use algebraic manipulations Partial Fraction Decomposition f 2x2 3xl5 dx x3 x2 l x 1 4 Use algebraic manipulations Partial Fraction Decomposition f 2x2 3x l 5 dx x3 x2 l x 1 2 Note 2x 3x l 5 2 3 3 x x2x 1 x l x21 4 Use algebraic manipulations Partial Fraction Decomposition 2x2 3x l 5 2 3 dx dx fx3 x2 l x 1 fx 1 x21 2 Note 2x 3x l 5 2 3 3 x x2x 1 x l x21 4 Use algebraic manipulations Partial Fraction Decomposition 2x2 3x l 5 2 3 dx dx fx3 x2 l x l fx l x2l 21nx l 3arctanxc 2 Note 2x 3x l 5 2 3 3 x x2x l x l x2l Called the Partial Fraction Decomposition 4 Use algebraic manipulations Partial Fraction Decomposition P Strategy for calculating the Partial Fraction Decomposition of a rational function QEX x 4 Use algebraic manipulations Partial Fraction Decomposition Strategy for calculating the Partial Fraction Decomposition of a rational function x Px Qx Want denominators of the form 2 2 x a and x a 4 Use algebraic manipulations Partial Fraction Decomposition P Strategy for calculating the Partial Fraction Decomposition of a rational function QEX x 1 When deg Px lt deg Qx 4 Use algebraic manipulations Partial Fraction Decomposition P Strategy for calculating the Partial Fraction Decomposition of a rational function QEX x 1 When deg Px lt deg Qx 0 Factor the denominator Qx 4 Use algebraic manipulations Partial Fraction Decomposition P Strategy for calculating the Partial Fraction Decomposition of a rational function QEX x 1 When deg Px lt deg Qx 0 Factor the denominator Qx o A factor of the form x c contributes Xc 4 Use algebraic manipulations Partial Fraction Decomposition Strategy for calculating the Partial Fraction Decomposition of a rational function x 1 When deg Px lt deg Qx 0 Factor the denominator Qx o A factor of the form x c contributes A x c n A 0 A factor of the form x c contributes A1 A2 4 Use algebraic manipulations Partial Fraction Decomposition Strategy for calculating the Partial Fraction Decomposition of a rational function x 1 When deg Px lt deg Qx 0 Factor the denominator Qx o A factor of the form x c contributes A x c n A o A factor ofthe form x c contributes A1 A2 2 n X C x c x c o A factor of the form x2 a2 contributes A 1 XCl 4 Use algebraic manipulations Partial Fraction Decomposition Strategy for calculating the Partial Fraction Decomposition of a rational function x 1 When deg Px lt deg Qx 0 Factor the denominator Qx o A factor of the form x c contributes A x c n A o A factor ofthe form x c contributes A1 A2 2 n X C x c x c o A factor of the form x2 a2 contributes A 1 x a 2 When deg Px 3 deg Qx do long division first and then apply rules above to the remainder term 4 Use algebraic manipulations Partial Fraction Decomposition Example xZ l 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 l xZ l x lxl 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 l A xZ l x l xl x l 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 1 A A x2 1x 1x0 x l xl 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 1 A1A2 xZ l x lxl x l xl How to find A1 and A2 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 1 A1A2 xZ l x lxl x l xl 1 A1xlA2x l J sumup xZ l x lxl 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 1 A1A2 xZ l x lxl x l xl Choose A1 and A2 so that numerators are identical 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 1 A1A2 xZ l x lxl x l xl 1 A1A2xA1 A2 x l 2 x l 2 A1A2O Al Azzl 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 1 A1A2 xZ l x lcp0 x J xl 1 A1A2xA1 A2 x l xZ l A1A2O 3 41212 A22 12 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 1 A1 A2 x2 1x 1x1 x lxl 1 A1A2xA1 AZ l A1A2O A1212 Al Azzl A22 12 2 2 x l x 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 l 1111 xZ l x1x1 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 l l l l l x2 1x 1x1 Em Izlcbzllwllw x l 2 x l 2 xl 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 l 1111 x2 1 x lxl ZEE EE x l 2 x l 2 xl l l lnx l lnxlc 2 2 4 Use algebraic manipulations Partial Fraction Decomposition Example 1 l 1111 x2 1 x lxl ZEE EE 21 arle de l de x l 2 x l 2 xl zllnx l llnxlc 2 2 1 x l ln c 4 Use algebraic manipulations Partial Fraction Decomposition Example x2 x 8 x3 2x2 3x 6 4 Use algebraic manipulations Partial Fraction Decomposition Example x2x 8 x2x 8 x32x23x6 x2x23 4 Use algebraic manipulations Partial Fraction Decomposition Example x2x 8 x2x 8 A1 A2xB2 x32x23x6x2x23 x2 x23 4 Use algebraic manipulations Partial Fraction Decomposition Example x2x 8 x2x 8 A1 A2xB2 x32x23x6x2x23 x2 x23 x2x8 A1A2x2 2A2Bzx3A1ZBZ x32x23x6 x2x23 sum up 4 Use algebraic manipulations Partial Fraction Decomposition Example x2x 8 x2x 8 A1 A2xB2 x32x23x6x2x23 x2 x23 x2x8 141Azx2 2A2Bzx3A1ZBZ x32x23x6 x32x23x6 Equate numerators A1 A2 l 2A2 B2 1 3A1 232 8 4 Use algebraic manipulations Partial Fraction Decomposition Example x2x 8 x2x 8 A1 A2xB2 x32x23x6x2x23 x2 x23 x2x8 141Azx2 2A2Bzx3A1ZBZ x32x23x6 x2x23 Equate numerators A1 A2 1 A1 2 2A2 B2 1 2 A2 1 3A1ZB2 2 8 B1 2 1 4 Use algebraic manipulations Partial Fraction Decomposition Example x2x 8 x2x 8 2 x l x32x23x6x2x23 x2x23 x2x8 141Azx2 2A2 2x3A1 282 x32x23x6 x2 23 Equate numerators A1A2 1 A1 2 2A2 B2 1 2 A2 1 3A1ZB2 2 8 B1 2 1 4 Use algebraic manipulations Partial Fraction Decomposition Example 2 I3x2x8dxj22ljdx x 2x 3x6 x2 x 3 4 Use algebraic manipulations Partial Fraction Decomposition Example 2 2 I3x2x8dxzj ljdx x 2x 3x6 x2 x 3 1 26 1 Z Z39lx2dx39lx23dx 4 Use algebraic manipulations Partial Fraction Decomposition Example 2 2 I3x2x8dxzj ljdx x 2x 3x6 x2 x 3 1 x l 2 2 d dx J x2 x J2623 x23j 4 Use algebraic manipulations Partial Fraction Decomposition Example 2 2 I3x2x8dxzj ljdx x 2x 3x6 x2 x 3 1 x l 2 2jx2dx39lx23dx Ix23dx 4 Use algebraic manipulations Partial Fraction Decomposition Example 2 I3x2x8dxzj2xljdx x 2x 3x6 x2 x 3 1 x l 2 2jx2dx39lx23dx Ix23dx 21nx2 wx2 4 Use algebraic manipulations Partial Fraction Decomposition Example 2 I3x2x8dxzj2xljdx x 2x 3x6 x2 x 3 1 x l 2 2jx2dx39lx23dx Ix23dx 21nx2llnx2 3 2 wx23 4 Use algebraic manipulations Partial Fraction Decomposition Example 2 I3x2x8dxzj2xljdx x 2x 3x6 x2 x 3 1 x l 2 2jx2dx39lx23dx Ix23dx 21nx2llnx2 3 2 l x f 2 2 dx arctan c x a a a 4 Use algebraic manipulations Partial Fraction Decomposition Example 2 I3x2x8dxzj2xljdx x 2x 3x6 x2 x 3 1 x l 2 2jx2dx39lx23dx Ix23dx 21nx2llnx2 3 iarctanijc 2 a a Chapter 7 Integration Sections 1 4 gt Calculation of antiderivatives integrals 1 Learn some fundamental properties of integrals 2 Use known differentiation formulas backwards 3 Use differentiation rules backwards chain rule substitution product rule integration by parts 4 Use algebraic manipulations eg partial fraction decomposition 5 Use tables of integrals 5 Use of Table of Integrals Refer to inside back cover of text 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx These come from integration by parts 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex COS x and sinx J sin 2x sin 3xdx 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 1 32 22 J sin 2x sin 3xdx 2 cos 2x sin 3x 2sin 2x cos 3x c 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 111 Products of Polynomial px with In x ex cos x sinx These come from integration by parts Note the recursive nature of the formulas 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 111 Products of Polynomial px with In x ex cos x sinx 33 2x 3 e xdx az l pxx2 2x3 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 111 Products of Polynomial px with In x ex cos x sinx 33 2x 3 e xdx iltx2 2x 3 e x a l pxx2 2x3 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 111 Products of Polynomial px with In x ex cos x sinx 33 2x 3 e xdx iltx2 2x 3 e x a l 1 px x2 2x 3 DZ 2x 2 e x 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 111 Products of Polynomial px with In x ex cos x sinx 33 2x 3 e xdx iltx2 2x 3 e x a l 1 px x2 2x 3 DZ 1 13 2x 2 e x 2 e x c 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 111 Products of Polynomial px with In x ex cos x sinx xz 2x3exdx x2 3ex c 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 111 Products of Polynomial px with In x ex cos x sinx IV Integer Powers of sinx and cosx These come from integration by parts Note the recursive nature of the formulas 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of excos x and sinx 111 Products of Polynomial px with In x ex cos x sinx IV Integer Powers of sinx and cosx l 4 1 J s1n4xdx2 Zs1n41xcosxT Sin4 2 x dx 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of excos x and sinx 111 Products of Polynomial px with In x ex cos x sinx IV Integer Powers of sinx and cosx l 3 J s1n4 xdx s1n3 xcosx J s1n2 xdx 4 4 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 111 Products of Polynomial px with In x ex cos x sinx IV Integer Powers of sinx and cosx l 3 J s1n4 xdx s1n3 xcosx J s1n2 xdx 4 4 l l J s1n2 xdx s1nxcosx J s1n0 xdx 2 2 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 111 Products of Polynomial px with In x ex cos x sinx IV Integer Powers of sinx and cosx l 3 J s1n4xdx2 s1n3xcosx J s1n2xdx 4 4 2 1 1 0 J s1n xdx s1nxcosx J s1n xdx 2 2 l l sinxcosx x 2 2 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 111 Products of Polynomial px with In x ex cos x sinx IV Integer Powers of sinx and cosx 4 1 3 3 1 1 J s1n xdx s1n xcosx s1nxcosx x 4 4 2 2 J s1n xdx s1nxcosx J s1n xdx 2 2 l l sinxcosx x 2 2 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of excos x and sinx 111 Products of Polynomial px with In x ex cos x sinx IV Integer Powers of sinx and cosx 4 1 3 3 3 J Sll l XdXZ ZSII I XCOSX SIHXCOSX XC 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions 11 Products of ex cos x and sinx 111 Products of Polynomial px with In x ex cos x sinx IV Integer Powers of sinx and cosx V Quadratic in the Denominator These are basic to integration by partial fraction decompositions 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions II Products of ex cos x and sinx III Products of Polynomial px with ln x ex cos x sinx IV Integer Powers of sinx and cosx V Quadratic in the Denominator VI Integrals Involving 612 x2 612 x2 c2 a2 a gt 0 These come from trigonometric substitutions 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions II Products of ex cos x and sinx III Products of Polynomial px with ln x ex cos x sinx IV Integer Powers of sinx and cosx V Quadratic in the Denominator VI Integrals Involving 612 x2 612 x2 c2 a2 a gt 0 de mg 1 ax 3x2 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions II Products of ex cos x and sinx III Products of Polynomial px with ln x ex cos x sinx IV Integer Powers of sinx and cosx V Quadratic in the Denominator VI Integrals Involving 612 x2 612 x2 c2 a2 a gt 0 Jxl3x2dxl V3x23j 1 0li 2 3x2 x3x2 1 J 3x2 dX ln 5 Use of Table of Integrals Refer to inside back cover of text I Basic Functions II Products of ex cos x and sinx III Products of Polynomial px with ln x ex cos x sinx IV Integer Powers of sinx and cosx V Quadratic in the Denominator VI Integrals Involving 612 x2 612 x2 c2 a2 a gt 0 Jx3x2 dx 3x2 3ln x3x2 c

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

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

#### "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!"

#### "I used the money I made selling my notes & study guides to pay for spring break in Olympia, Washington...which was Sweet!"

#### "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!"

#### "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."

### 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

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.