AntiTrnn (Antiderivtive) AntiTrnn dri seh fngsi f dl seh fngsi F sedemiin hingg F = f Pernytn: Integrl T Tent f dic integrl t tent dri f terhdp, Artiny dl mendptn sem ntitrnn dri f. E. AntiTrnn dri f = 6 dl F = + rn F = f. Tnd Integrl f Integrnd diset peh integrsi Konstnt dri Integrsi Setip ntitrnn F dri f hrs dlm ent F() = G() + C, dimn C dl seh onstnt. Atrn Pngt dri Integrl TTent, Bgin I n+ n if n n + Perhtin 6 = + C E. Mewili sem ntitrnn yng mngin dri 6.
Atrn Pngt dri Integrl TTent, Bgin II = = ln + C Integrl TTent dri e dn e = e + C ln Atrn Jmlh dn Krng f ± g = f ± g f = f ( constnt) E. + = + = + + C Atrn Perlin dengn Konstn E. = = + C Contoh: E. Dptn integrl t tent dri: 7 e d + 6 = e d 7 d + d 6d = e 7ln = 6 + C Posisi, Keceptn, Perceptn Ji s = s(t) dl fngsi posisi dri seh oye pd st wt t, m ds dv Keceptn = v = Perceptn = = dt dt Bent Integrl s( t) = v( t) dt v( t) ( t) dt =
Integrsi dengn Sstitsi Metode integrsi yng erhngn dengn trn rnti. Ji dl fngsi dlm, m it is mengnn forml/persmn f f = d d / Integrsi dengn Sstitsi E. Dptn integrl: ( ) 9 9 d = 0 Amil = +, m d = d 0 + 0 0 Stitsi Integrln Sstitsi lng E. Dptn d Let 7 then 0 7 = = / 7 = d 0 / = + C 0 / ( 7 ) / Tentn, dptn d Sstitsi Integrln Sstitsi E. Dptn Let ( ln ) = ln then d = ( ln ) = d ( ln )
t E. Dptn e dt t e + t d Let = e + then = dt t e t e dt = d t e + ln t ln ( e + ) Espresi Integrl yng mengndng + Atrn ( + ) n+ ( ) n + n ( n + ) ln + = + + C + + e = e + C + + c = c + C ln c Jmlhn Riemnn Ji f dl sh fngsi yg ontin, m jmlhn Riemnn dri n gin yng sm nt f sepnjng selng [, ] didefinisin sg: n = 0 f = f ( ) + f ( ) +... + f ( ) 0 n [... ] = f + f + + f 0 n dimn = 0 < < K < n = dl gin = ( ) / n Integrl Tent Ji f dl fngsi yg ontin, integrl tent f dri e didefinisin sg n n = 0 f = lim f fngsi f diset integrnd, ng dn diset limits dri integrsi, dn peh diset peh dri integrtion.
Pendetn Integrl Tent E. Hitng jmlhn Riemnn t integrl menggnn n = 0. 0 n 9 f = = 0 = 0 = (/ ) ( / )... (9 / ) + + + (/ ) =.8 Integrl Tent f dic integrl dri e dri f(). Peh is dirh menjdi peh p sj, contoh f = f ( t) dt Integrl Tent Segi Totl Ji r() dl tingt perhn dri qntity Q (dlm nits Q per nit ), m totl t mlsi perhn dri qntity st erh dri e dierin oleh Totl chnge in qntity Q = r Integrl Tent Segi Totl E. Ji pd st t menit nd erpergin dengn lj per-meter per-menit seesr v(t), m jr totl yng ditemph dri menit e- smpi menit e-0 dierin oleh 0 Totl chnge in distnce = v ( t ) dt
Are diwh Krv Ler: = n (n persegi pnjng.) Ide: Mendptn re seenrny (tept/persis) diwh rv sh fngsi. Metode: Menggnn t hingg persegipnjng dgn ler yg sm dn menghitng re dgn limit. y = f Memperirn Are Perirn re diwh rv Menggnn n =. [ ] f = on 0, [ ] A f + f + f + f 0 A f ( 0) f f ( ) f + + + 9 7 A 0 + + + = Are Diwh Krv y = f Interpretsi Geometric (Sem Fngsi) f ontin, tnegtif pd [, ]. Are dl n n = 0 Are = lim f = f ( ) Are R AreR + Are R f ( ) = R R R y = f 6
Are Menggnn Geometry E. Gnn geometry nt menghitng integrl Are = ( ) Are = = = Teorem Dsr Klls Ji f dl fngsi yg ontin pd [, ].. If A = f ( t) dt, then A = f.. Ji F dl serng ntitrnn yng ontin dri f dn is didefinisin pd [, ], m f ( ) = F ( ) F ( ) Teorem Dsr Klls E. If A = t + tdt, find A. A = + Mengevlsi Integrl Tent E. Hitng + + = ln + ( ( ln ) ( ln ) = + + = 8 ln 6.906 7
Sstitsi nt Integrl Tent / E. Hitng ( + ) 0 let = + d then = / / + = d 0 0 = / 6 = Perhtin hw limit integrsi erh 0 Menghitng Are E. Dptn re yg ditsi oleh sm, gris vertil = 0, = dn rv y =. 0 dl t negtif pd [0, ]. = 0 ( ) ( 0 ) 0 Antitrnn = = 8 Teorem Dsr Klls 8