EXERCÍCIOS DERIVADAS E INTEGRAIS_260324_115535

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Exercicios de reviscio de derivada de V1 2 1) Derive as a) f(x) = b) fix) = e = e) = -e-x d) = e) fix) = f) = etgx mostre que as sai da diferencial 3) mostre que y= + satisfaz y" + 2y'- 4) f" se = 1. = = Zlax X 1/2 d) = xh10 y= lny=xhx y um J'= X e= y'= 3x2 y x x y x y V=X 1+x 6- = 7. = f(x) J-x y'x = y(1-x) y 3.9. X = e 2 du= 1/xdx = ( du= dv= dx V=X du=2xdx 2 ) Z = 2 1 2 7. V-X lnx 2. lnx X 2 = x) C 1. 2. 3- 2 + du=2xdx du=dx 2. 3 4Be (+) =4. du= 7 2 -2 4 -4 1. Sxdx U=Sx Sx + C 2- = Sx+1 3. JJx dx 2 1, 2 3 3 4. S It dt J/t th S. dx = J = conec 6- S = ) = J C 2 t 2 do = 0 do tgo C 8. do do dx = = 2 e = e 2xdx + = 2. dx dx =7 dx X Ver depois 2xdx USI + 2 102 Sx dx du=dx ! 3u du =2 203/2 ( S du=2xdx 2 ) 2 dx = = + S dx dx ex dx = ex 7) = dx = dx = S du u = = dx) = = = ln Varideo 3 técnica10) S + + dx = u = In x 2x / + dx. (2 dx. u=x+tgx x + 11) S tg3 x see dx = u du = u= lnx =7 du= dx x = = = w= = u => dw= = = ( 1+ dw = 4 3 + w8 8 = + + = 4 + (Inx) + 4 3 8 3 10a)d d Rx dx 2 b) d d UV= dx dx S): X 3/2 3/2 b) X = dx Z d) dx Lx dx I 3 x dx + = du=dx b dx I pono Identidade ! que, en fizer por exemplo, = e vice dx = S = cotgo = X + C 2 2) X 3 C + dx tgxdx + +e) S cosx dx J = conec + ( f) dt +2 dt It = t -1 S = ( = 3 3 3 3 3 (x) (-2,81 6) Para quais valores de aeb a reta 2 y = b e' tangente à paralola = em = da parabola ma derivada Zax Derivada da parabola I } a por na 4 = 2 a) f(x) = X e derivar isso: 2 1+2x v'=2 1+2x v² du=4x3 b) tg Tenho uma derivação de uma função dentro de uma dentro de uma derivando: y == a nec ax c) ) = Chamo x 2 ( arcty ( = ( arcty 2 Sei que = X derivada dc 2 y'=- 1 2 Derivada de 2 , 2 X 10) Determine a y= ca h(x) -6-4 - 5/2 dy dx = senx l = 1 X 4 4 4 dy = 1 x e = 5 c) f"(x) = 9. = + = cor 2 4 2 2 2 U=2x dx 2 2 b) dy 1 S ) dx -7 -log C.S dx X c) = + = C=1 2X=U S 2 X + dx 6 0 = 6 6 x²+b, X + 63 3 B + + 3 3 3 2x t + 2 + C=3 Integral V V (A) x dx integral (1) + (8) S ( + 1/1+x2) dx (2) + (c) S ( seno + sen do (D) + ) dx (3) + x + x + K (E) dx (4) 1/3 (x3 3 +K (5) -coso + K (F) (2x) 1/senx dx (6) - 1/x dx X X dx = -2.x 1/2 x3/3 X 1/2 = Se ao 4 3 B S dx X+ dx 1, x3, X+ X + Se relaciona 3 3 3 of dodo =7 = - du = 3 C C U+C = ( 3 33 D- X + X dx = S X X X X 3 X E. S dx 2 + C 2. 3x 2+1) Jx 6 du= 3 du 3 ex3 + C 4. t C S. 2 2 2 du 4 du 4 4 10 1 6 + S 2/13 6. t t dt -du= Ldte dx du= dx C do 2 2 du = 2.) =-2 9.) 2 f 5x ) dx 3x 3d 1 3 ) dx du=3x2dx 3 du 3 30 3 = 4 4 5/4 =) S/u + 39 S 27 5 => du=3dx dx=do 3 3 = 3 13. > 2 - Judu 1/50 Su du) 23 1.2 1.2 xdx 2 3 "du= = dx S Judx = = f C S dt, 2 2du=1 dt J 8. do 7 7 =7 6 20 = 2 = 2 C = - = 2 21- S dx U=3x+1 du=3dx - , du = 3 = 3/2 3 33 g (0,1) + = g dx dx V=X -) = = 2 2. S X3, dx x²=w W=V du=dw ( 4. X X=U du-dx =) v) Jxdx x=udu=dx v=tgx xtgx- tgx= = dx corx U S. S 3 25x dx 3 3 (1+x) 3 (1+x) 3 U 3 3 ( 3 6. -> du= 2x dv=dx. dx dx dx) I 7. St = = ( dt) 2 4 = 2 2 2 28. / dx I x ) du ) X ( 2/ S te dt_, at at e te V= e e a a a 3 X dx dx = 2 2 2 3 3x C 2 4 S ( S I II T du= dr=tdt V= 12/2 = 2 +2/2 dt 2 2 + 2 4 = C 2 12 = e U=tgx us S 3 3 13. S I - xdx 2 Y+ = 14.5 dx du= dx V=-X⁻¹ dx X X X Jxdx du=xdx Z X 2 2 2 X 2 IS. xlnx-x = coix free3dx - 7 2 2 17 tgx+ > S tg'xdx tg2x 19 522 2 2 L du=dx calx=dv V: ( 2 2 2 4 4 4 8 = 2 ln = 2 = = v=tyx 4 I + = 2 4 8 8 2 )3 C 2 4 3 dx du 3 S 4 dx dx =6 - S 7 dx= U6 du C + 3 8 tgx =7 2 C dx = 1/2 guadrado: 1/4 -7 4 4 2 dx 8 16 f ( 10 - => du 08 S 7 11 E otipo de que necai 4 1/2 2 4 4xdx = 4x 4x 4 8 6 12 Se C 2 13 1/4 dx 8 8 8 1. 2 6 2 2 8Res= X ( 2 16 64 48 14 dx : = S 3 coix dx us du = 2 + 4 C 6 = S 2 2x 2 X dx 4 4 8 + C 4 7 co2xdx X=V du=dx w3/3 3 1 3 3 -> + " - ( 3 3 8 X = 0 1) Calcule a) are sen (52/2) ; b) arcsen (-1) ; c) (0) are sen (-1/2) ; ; f) g) are sec (2) ; h) sec (-2) ; i) are tg tg sen (-2)) ; l)sen1. IT 1/2 2 4 coz I V3 Z go 6 1 -II 2 IT 2 6 53 = IT 3 f' -II 6 g) II 3 -II 3 S3 11 3 4 3 2 2 = 53 - 2a (or a 2 3 a tg 3 3 - 17a 4a = 4 = 4517 no negative) a 17 2 20 = + - que 2 = X 2 X X (2x- X X b) X 1, = 2 lna) x It 2_1 X bl caty conec X X X 2 X3 dx i) iii) subs trig. ii) subs J a) S X 2 X V 3 X dx = 3/2 = 3/2 - - - 2 3 3 3/2 5/2 du - 3/2 + 3 3 IS 3 IS 5/2 b) X 3 dx U du = du= Z 1/2 2 2 12 U 5/2 3/2 = + 25 3 S 3 c) S X corodo 2 C 3/2 S 3 3dy i) 2 du ii) 4 y do S C 4 du=8tdt tdt= du du= - - 8 8 4 4 ii) dt = recotgodo 2 4 = 4 c) X dx U 2xdx du =xdx du + Z 2 2 I 4 { " 0 " 2 = R = 2 1, Res= 2 dx 3 9-x2 X = = C 3 S X dx xdx=du - C + -2 9-x2 2 U 5 3- X dx du ( 2 C - xdx= du/2 24- dx 2 = 4-x2 4 tgo.C do = 4-x2 = X S. dx 10 = S/x = Suco = 2-25 S C dx dx= = 4 4) 2 i " 2 N) 2 7- dx no = = dx: conec S + + 8- dx x tgo=x = X = seco do S tgo S do tgo tgo do du tS du 2 U-1 C U 2 / C ln + C dx 3 = 3 27 = g S 1, = C 2 2 2 4 = 9 1 C = C Bx Bx 2 2 10. dx 3 5x 3/ = Sx 25 Scorodo 31 25 9-25x2 = do 9 > " 3 125 125 2 4 2 2 125 250 50 4 X dx 10 16 = 64.16) = 3 S 1024 ( 3 S - 1024 ( 4 4 ) 16 5/2 + ( = 3 S 3 S dx X to 2( - do = 2 x² I = = do do - du= I 4 coro 4 1 4 = ( 4 4xB- dx X 10 S tgo = du= = 25 ( = = + ( Agora -7 Substituin 2 - dx dx 10 -3+4 corodo +1 do C ( 2 = (x-2) 2 2 2 du= dx U 2x-1 tgo= = S xdx = dx= 2 / = 4 reco do no corodo 4 f 4 + ( or = U = U T 1, C 4 4 C 4 2 16- dx X = 0 It Cdu du= = do 10 In = ( 18- dx X 10 + = 19 y 10 S y3 dy= do 10 + S S 2 10 10 S dx = = X lax et do dt = / C +9 22) x dx de 2L S dt du 0 3 du= du=etdt -3) do ln In C 3 dx X dx 2 2 4 = 2 + 2 2Lista - 2026 as integrais indefinidas 1) S dx (x (x+2)3 13) (x 3x+3 dx (x+1)3 3x2 dx 14) S x dx 3) S + 10x2 dx + 10 15) S 1+x3 dx 4) S dx 16) S are dx 5) S coso do -5 6) S x²+x. dx to 7) S dx + 8)f dx x²+x + 9) S x dx / + 10) t dx + 2x 11) e4x dx 12) S do (tgo)3 II) Le tal que = 1 e S f(x) dx e'uma funcho emontreIII) Leja of uma racional tal que f(x) = + 12x 8 . Determine a f, (x-2)2 2 sabendo que1. dx s X=U-2 du 4 C t 2. S 3x2 du=dx 3 du 3 6 3 C 3. dx x4-6x3-10x2 X dx x3/3 X dr U-3 3 du (v-1) 1/wdw C 2 3 4. dx 3x2-1 dx A B C no Bx B Cx2 122 3x²-1 + 3 x (=2 A- X no + C B=-1 S. do du= coodo (U+SI du du A B t Alu no as Se U= " - B(6)=1 B=1/6 A= 1/6 S 1 S du I 6 6 U-1 6 6 6 6. dx dx (x-2)(x+2)(x²+1) X-2 x+2 3y-4 Soma prod 4,-1A X A Ax- Bx3-2 + Bx-2B+Cx + 4D = -2A-2B+D=1 SD = A+B-4C=1 PI B=3/10 = -1 3 dx 1/5 dx t 10 10 (x-21 10 10 1 dx I 1x X 3 C = S 10 10 10 7 S dx dx X-1 U du du V dx=du/2 4 4 4 8 YES dw=2udu 8. X dx R X dx dt 2 dt + dx:2tdt 2 S dx X dx 3/2 3/2 4 11 -> N v du dw = 4 2 do+ C " 4 4x+2 S dx U= U du dx 2 U du 2 A B ЧА = B=4/3 6 4-30 6 3 3A D + 3 2 S du du= dw/3 g 2 S dw 9 2 do no du du C du du = + = -B=1 U U U B dx A.B.C B=-1 (x+1) (x+113 2A+B=3 C=2 dx S / du / C 2 14. S X dx Zodu dodt 4 S t²-1 dt 4 4 +1 A.B = -B-A t-1 A-B=1 = 2A= = 2 2 = 2 ln ++1 + ( 4 In + + C IS S X dx 3 2 1/2 2 dx ? anoty + X 3 d X., 1dx no dx X X X Bx+C { X