3 2x. 3x 2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
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1 Math B Intgration Rviw (Solutions) Do ths intgrals. Solutions ar postd at th wbsit blow. If you hav troubl with thm, sk hlp immdiatly! () 8 d () 5 d () d () sin d (5) d (6) cos d (7) d Prpard by Vinc Zaccon For Campus Larning Assistanc Srvics at UCSB
2 Math B Intgration Rviw (Solutions) () u 8 u u u d 8 du 8 du du C 8d du 8 d W nd to us a substitution to dal with th 8, and thn rwrit th root symbol as a fraction powr. This substitution always works th sam way if you hav somthing lik k, you just nd up dividing by k. 8 C Prpard by Vinc Zaccon For Campus Larning Assistanc Srvics at UCSB
3 Math B Intgration Rviw (Solutions) () u du u u 5 C C d 5 5 u du du d Hr you nd to rcogniz that th drivativ of th stuff in th squar root is actly what you hav on top. This maks th substitution work smoothly. Prpard by Vinc Zaccon For Campus Larning Assistanc Srvics at UCSB
4 Math B Intgration Rviw (Solutions) () u du d u du u C C d du d Again, th drivativ of th stuff in th ponntial is multiplid in front, so th substitution will work nicly. Prpard by Vinc Zaccon For Campus Larning Assistanc Srvics at UCSB
5 Math B Intgration Rviw (Solutions) () sin d This on uss Intgration by Parts u dv sin d u dv uv v du du d v cos sin d cos cos d cos sin C B spcially carful with ngativ signs in ths typ. Prpard by Vinc Zaccon For Campus Larning Assistanc Srvics at UCSB
6 (5) d Math B Intgration Rviw (Solutions) W nd intgration by parts for this on, and w ll nd to do it tims (bcaus of th ). For ths rpatd ons I lik to us a shortcut so w don t hav to writ down th intrmdiat solutions at ach stp. Mak a tabl with th u and th dv, thn tak drivativs of th u sid until you gt to zro, and intgrat on th dv sid. Thn w can just writ down th final solution, rmmbring to altrnat signs. Try this on th long way first, and prov to yourslf why th shortcut works. u dv This shortcut works bcaus th function w us for th u part is a polynomial, which will go to zro vntually whn w tak nough drivativs. As long as th intgrals ar simpl this trick savs you lots of tim. It will work bst for things lik n a d or n sin(a) d d 8 C Prpard by Vinc Zaccon For Campus Larning Assistanc Srvics at UCSB
7 Math B Intgration Rviw (Solutions) (6) cos d W us intgration by parts twic for this on, but it s a bit tricky. Mak sur you writ down both sids of th quation at ach stp it will b much asir to rmmbr what to do at th nd. Aftr th first applictaion of intgration by parts it looks lik w havn t gottn anywhr, but w can pand th nw intgral using intgration by parts again. w hav com full circl to th intgral w startd with. Add it to th othr sid and divid by to gt th final answr. Don t forgt to tack on th constant of intgration. Prpard by Vinc Zaccon For Campus Larning Assistanc Srvics at UCSB
8 Math B Intgration Rviw (Solutions) (7) d W could try a substitution trick lik in intgral (), but it dosn t quit work out. Start by factoring th dnominator: ( )( ) d W will us a partial fraction dcomposition to turn this into sparat fractions, ach of which will b asily intgratd. This is just an algbra trick. Don t panic. ( )( ) ( )( ) A( ) B( ) 0 (A B) (A B) A B A ;B ( )( ) A A( ) B( ) ( )( ) 0 A B B St th numrators qual Collct lik trms and solv th rsulting systm Now w can do th intgrals sparatly. In this cas w gt logarithms. ( )( ) d d ln( ) ln( ) C Prpard by Vinc Zaccon For Campus Larning Assistanc Srvics at UCSB
u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
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