Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx
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1 Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f. * width Δ nd hoose i from eh intervl. Indefinite Integrl : f d = F * Then f ( ) d = lim f ( i ) Δ. where F( ) is n nti-derivtive of f ( ). n i = Fundmentl Theorem of Clulus f is ontinuous on [, ] then Vrints of Prt I : d u g = f ( t) dt is lso ontinuous on [, ] f () t dt u f u d = d d g = f t dt = f d. f () t dt v f v d = v f is ontinuous on[, ], F( ) is d u f () t dt u f u v f v d = v F = f d) Prt I : If nd () Prt II : n nti-derivtive of f ( )(i.e. then = f d F F. ± = ± ± = ± f g d f d g d f g d f d g d 0 f d= = f d f d If f g on then If f 0 on then 0 Properties f d = f d [ ] [ ], is onstnt f d = f d, is onstnt = f d g d f d f d f t dt f d f d If m f M on then m ( ) f( d ) M( ) n n d= n d =, n d= d= ln d = ln ln udu= uln ( u) u u u e du e = Common Integrls osudu= sin u sin udu= osu se udu= tn u seutn udu = seu su ot udu = su s udu= ot u tn udu= ln seu seudu= ln seu tn u u du = tn u du u = sin u Visit for omplete set of Clulus notes. 005 Pul Dwins
2 Clulus Chet Sheet Stndrd Integrtion Tehniques Note tht t mny shools ll ut the Sustitution Rule tend to e tught in Clulus II lss. u Sustitution : The sustitution u = g will onvert du = g d. For indefinite integrls drop the limits of integrtion. E. 5 os d u = du = d d = du = u = = :: = u = = 8 Integrtion y Prts : udv= uv vdu nd g f ( g ) g d= f ( u) du using g = = sin ( u) = ( sin ( 8) sin ) 8 5 integrl nd ompute du y differentiting u nd ompute v using v E. d u = dv = e du = d v =e e d= e e d=e e 5 os d os u d udv = uv vdu. Choose u nd dv from E. 5 ln d = dv. u = ln dv = d du = d v = ln d= ln d ( ln ) = = 5ln 5 ln Produts nd (some) Quotients of Trig Funtions n m n m For sin os d we hve the following : For tn se d we hve the following :. n odd. Strip sine out nd onvert rest to osines using sin = os, then use the sustitution u = os.. m odd. Strip osine out nd onvert rest to sines using os = sin, then use the sustitution u = sin.. n nd m oth odd. Use either. or. 4. n nd m oth even. Use doule ngle nd/or hlf ngle formuls to redue the integrl into form tht n e integrted.. n odd. Strip tngent nd sent out nd onvert the rest to sents using tn = se, then use the sustitution u = se.. m even. Strip sents out nd onvert rest to tngents using se = tn, then use the sustitution u = tn.. n odd nd m even. Use either. or. 4. n even nd m odd. Eh integrl will e delt with differently. sin sin os os os sin = os Trig Formuls : =, = ( ), ( ) E. tn se 5 d = 4 ( se ) se tn se d 4 ( u ) u du ( u se) 5 4 tn se tn se tn se d d = = = = se se E. sin5 d os 5 4 sin sin sin (sin ) sin = = os os os (os ) sin = = os ( u) u u4 = du = u u d d d ( os ) d u du = se ln os os Visit for omplete set of Clulus notes. 005 Pul Dwins
3 Clulus Chet Sheet Trig Sustitutions : If the integrl ontins the following root use the given sustitution nd formul to onvert into n integrl involving trig funtions. = sinθ os θ = sin θ = seθ tn θ = se θ = tnθ se θ = tn θ E. 6 = θ d= osθ dθ d 49 sin 4 4sin 4os 4 9 = θ = θ = os θ Rell =. Beuse we hve n indefinite integrl we ll ssume positive nd drop solute vlue rs. If we hd definite integrl we d need to ompute θ s nd remove solute vlue rs sed on tht nd, if 0 = if < 0 In this se we hve 4 9 = osθ. 6 d = 4 sin θ θ sin θ 9 ( os ) ( os ) θ θ dθ = s dθ = otθ Use Right Tringle Trig to go to s. From sustitution we hve sinθ = so, From this we see tht 49 otθ =. So, d = Prtil Frtions : If integrting P d where the degree of Q P is smller thn the degree of Q( ). Ftor denomintor s ompletely s possile nd find the prtil frtion deomposition of the rtionl epression. Integrte the prtil frtion deomposition (P.F.D.). For eh ftor in the denomintor we get term(s) in the deomposition ording to the following tle. Ftor in Q( ) Term in P.F.D Ftor in Q A A B ( ) ( ) Term in P.F.D A A A A B A B 7 d ( )( 4) E d = ( )( 4) 4 = d d ( ) = 4ln ln 4 8tn Here is prtil frtion form nd reomined. A ( A B C 4) ( B C ) ( ) = = Set numertors equl nd ollet lie terms. 7 = A B C B 4A C Set oeffiients equl to get system nd solve to get onstnts. A B = 7 C B = 4A C = 0 A= 4 B = C = 6 An lternte method tht sometimes wors to find onstnts. Strt with setting numertors equl in previous emple : 7 = A( 4) ( B C) ( ). Chose nie vlues of nd plug in. For emple if = we get 0 = 5A whih gives A = 4. This won t lwys wor esily. Visit for omplete set of Clulus notes. 005 Pul Dwins
4 f Clulus Chet Sheet Applitions of Integrls Net Are : ( ) d represents the net re etween f nd the -is with re ove -is positive nd re elow -is negtive. Are Between Curves : The generl formuls for the two min ses for eh re, d y = f A= d = f y A= upper funtion lower funtion & left funtion right funtion If the urves interset then the re of eh portion must e found individully. Here re some sethes of ouple possile situtions nd formuls for ouple of possile ses. A= f ( y) g( y) = A= f g d d A f g d g f d Volumes of Revolution : The two min formuls re V = A d nd V = A y. Here is some generl informtion out eh method of omputing nd some emples. Rings Cylinders A = π ( outer rdius) ( inner rdius) A = π ( rdius) ( width / height) Limits: /y of right/ot ring to /y of left/top ring f, f y, Horz. Ais use g( ), A( ) nd d. Vert. Ais use g( y ), A( y ) nd. Limits : /y of inner yl. to /y of outer yl. f y, f, Horz. Ais use g( y ), A( y ) nd. Vert. Ais use g( ), A( ) nd d. E. Ais : y = > 0 E. Ais : y = 0 E. Ais : y = > 0 E. Ais : y = 0 outer rdius : f inner rdius : g outer rdius: g inner rdius: f rdius : y width : f ( y) g( y) rdius : y width : f ( y) g( y) These re only few ses for horizontl is of rottion. If is of rottion is the -is use the y = 0 se with = 0. For vertil is of rottion ( = > 0 nd = 0 ) interhnge nd y to get pproprite formuls. Visit for omplete set of Clulus notes. 005 Pul Dwins
5 Wor : If fore of F moves n ojet in, the wor done is W = Clulus Chet Sheet F d Averge Funtion Vlue : The verge vlue of f ( ) on is fvg = f d Ar Length Surfe Are : Note tht this is often Cl II topi. The three si formuls re, L= ds SA = π y ds (rotte out -is) SA = π ds (rotte out y-is) where ds is dependent upon the form of the funtion eing wored with s follows. ( d ) d ds = d if y = f, ds = if = f y, y d () () ds = dt if = f t, y = g t, t dt dt dr ds = r dθ if r = f θ, θ dθ With surfe re you my hve to sustitute in for the or y depending on your hoie of ds to mth the differentil in the ds. With prmetri nd polr you will lwys need to sustitute. Improper Integrl An improper integrl is n integrl with one or more infinite limits nd/or disontinuous integrnds. Integrl is lled onvergent if the limit eists nd hs finite vlue nd divergent if the limit doesn t eist or hs infinite vlue. This is typilly Cl II topi. Infinite Limit f d f d t. lim = t t. = lim t. = f d f d f d f d f d provided BOTH integrls re onvergent. Disontinuous Integrnd t. Disont. t : f d= lim f d. Disont. t : f d = lim f d. Disontinuity t t t t f d f d f d provided oth re onvergent. < < : = Comprison Test for Improper Integrls : If f g 0 on [, ) then,. If onv. then onv.. If divg. then f d Useful ft : If > 0 then For given integrl f ( ) d divide [, ] g d g d p d onverges if p > nd diverges for p. f d divg. Approimting Definite Integrls nd n (must e even for Simpson s Rule) define Δ = n nd into n suintervls [, ], [, ],, [ ] 0 with 0 n, n * * * Midpoint Rule : Δ ( n ) = nd n = then, f d f f f *, i i, i Δ f d f 0 f f f n f n Δ f d f 0 4f f f 4f f is midpoint [ ] Trpezoid Rule : Simpson s Rule : n n n Visit for omplete set of Clulus notes. 005 Pul Dwins
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