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# Integral

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### Integral

1. 1. INTEGRALINTEGRAL TAK TENTUINTEGRAL TERTENTU
2. 2. INTEGRAL TAK TENTU CONTOH : 1.∫ 3 dx = 3x + c 2.∫ 5 dt = 5t + c 3.∫ 8 dQ = 8Q + c4.∫ 56 du = 56 u + c
3. 3. 2. ∫ ax b dx = a x b+1 + c b+1CONTOH :1.∫ 4X3 dx = 4 x 4 + c = x4 + c 42. ∫ 3x8 dx = 3 x 9 + c =1/3X9 + C 9
4. 4. 3. ∫ aUb dU = a U b+1 + c b+1 U=f(x)CONTOH :1. ∫ (2X+ 1)dx = … 2. ∫ (4X + 4) dX = … -1 X2 + X (4X2+8X+6)3 4 (4x2+8x+6)2Jawab : jawab :Misal : U = X2 + X Misal : U =4X2+8X+6 dU =( 2X + 1)dX dU =(8X+8)dX ∫ (2X + 1)dx = ∫ dU dU =2(4X+4)dX X2 + X U dU =(4X+4)dX = Ln U + C 2 = Ln ( X2 + X ) + C ∫ dU = ∫ ½ U -3 dU 2U3 = ½.1/-2 .U-2 + C = - ¼(4X2+8X+6) -2 + C
5. 5. 4.∫UdV = U.V - ∫VdU RUMUS DI ATAS ADALAHCONTOH : RUMUS INTEGRAL PARSIAL∫X.eX dx = ….Misal : U = X du = dx dv = eX dx V=∫eX dX = eX + C∫X.eX dx = U.V - ∫V dU = X.eX - ∫ eX dx = X.eX - eX + C
6. 6. 5.∫ ex dx = ex + c6.∫[f(x) + g(x)] dx =∫ f(x)dx+∫g(x)dx 7.∫n.f(x)dx = n∫f(x)dx
7. 7. SOALSELESAIKANLAH !1. ∫ X3 dX = … 6. ∫ √ 2 + 5X dX = …2. ∫X -4 dX = … 7.∫ (X2 + 3X + 4)3(2X + 3)dx =…3. ∫9X2 dX = … 8. ∫ X2 + 3X – 2 dX = …4. ∫5/X dX = … X5. ∫(X2 -√X + 4) dX = … 9. ∫X.e x² dX = …
8. 8. INTEGRAL TERTENTUUNTUK a < c < b,berlaku b b b b1.∫ f(x) dx = [F(X)] = F(b)- F(a) 4. ∫ k f(x) dx =k ∫ f(x) dx a a a a a b b b 2.∫ f(x) dx = 0 5. ∫ [f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx a a a a b a c b b3.∫ f(x) dx = - ∫ f(x) dx 6. ∫f(x)dx + ∫f(x)dx = ∫ f(x)dx a b a c a
9. 9. SOAL 6 01.∫ X dX = …. 5. ∫ (X2 – 2X + 3) dX = … 4 3 3 32. ∫ (X2 – 2X + 3 ) dX = … 6. ∫ (2X + 1)(3 – X) dX = … 0 1 1 43. ∫ (2X + 5) dX = … 7. ∫ ( √ X – X )2 dX = … -1 1 -4 84. ∫ (3X2 + 2X) dX = … 8. ∫ (X1/3 – X-1/3) dX = ….. -6 1 2 2a9. ∫ (X + 9X3) dX = … 10. ∫ (a + X ) dX = … 1 a
10. 10. BYAMIRULSYAH,MSi
11. 11. SURPLUS KONSUMEN Fungsi demand Fungsi demand SK SK SKP1 Q Q O Q1 O P P
12. 12. SURPLUS PRODUSEN P P SP P1 Fungsi supply SPP1 Fungsi supply Q QO Q1 O Q1
13. 13. P P Fungsi demand SK SK Fungsi supply P1P1 SP SP Q O Q1 O Q 0 Q1
14. 14. PENGETAHUAN DASARLUAS DAERAH Y CARA I : L= axt 2 5 L= 4x3 LUAS = …? 2 L= 6 satuan luas 2 X CARA II : Integral O 4 4 L= ∫(5-3/4x)dx – 2x4 CARA III: INTEGRAL 0 5 4 L=∫( 2 ) dy ² = (5X – ¾.1/2X )] - 8 0 Y= 5-3/4x = (5.4 – 3/8.16) – (5.0-1/4.0) – 8 X= 20/3 – 4y = (20 – 6) – 0 – 8 5 = 14 - 8 L = ∫ (20/3 – 4/3Y)dy = 6 satuan luas 2 L= 6 satuan luas
15. 15. LUAS DAERAH P 6 CARA I: INTEGRAL 5 LUAS P= 6 – 3/25 Q ² L=∫ ( 6 – 3/25Q²)dQ – 3x5 3 0 5 0 Q 5 L = (6Q – 3/25.1/3Q³)] – 15 0 L = 10 satuan luas CARA II: INTEGRAL 6 L=∫ (50 – 25/3P)1/2 dP 3 6 L = { 2/3(50 – 25/3P)3/2.(-3/25)} ] 3 L = { - 2/5 (50 – 25/3P)3/2 L = 10 satuan luas
16. 16. P LUAS= …? 2 P 2 Q 6 3 CARA II : INTEGRAL 6 2 ∫ L = 6X6 - (2 + 2/3Q)dQ Q 0 6 0 6 { L = 36 – 2Q + 2/3.1/2Q² }] 0CARA I : RUMUS L = 36 – 24 = 12 satuan luasL = axt 2 CARA III : integralL= 4x6 6 2 ∫ L = ( 3/2 P – 3 ) dPL = 12 satuan luas 2 6 L = ( 3/4P – 3P ) ] = 9 + 3 = 12 satuan luas 2
17. 17. LUAS DAERAH P P = 2 + 1/5Q²7 CARA I : INTEGRAL LUAS 52 L = 7x5 - ∫( 2 + 1/5Q²)dQ Q 0 50 5 ] L = 35 - (2Q + 1/5.1/3Q³) 0 L = 35 - 10 - 8 1/3 CARA II : INTEGRAL L = 16 ⅔ satuan luas 7 ∫ L = (5P - 10)1/2 dP 2 7 L = { 2/3(5P - 10) 3/2. ⅕ }] 2 L = 2/15.{ 25 } 3/2 L = 16 ⅔ satuan luas
18. 18. P P = 5 + 1/12Q21.Fungsi pendapatan 2. 12 dari suatu pabrik diberikan sebagai berikut : LUAS I 8 R = 6 + 350Q – 2Q2 LUAS IIFungsi produksinya : P = 12 - 1/9Q2 Q = 3L 5Jika jumlah tenaga Q kerja yang ada 25 0 6 orang,berapakah MPRL dan jelaskan artinya .
19. 19. 6 P P = 5 + 1/12Q2Luas I = ∫(12 - 1/9Q2)dQ - 8X6 2. 12 0 6= ( 12Q + 1/9.1/3Q3) ] - 48 0 LUAS I= (12.6 + 1/27.63 – (12.0 + 8 1/27.03) - 48 LUAS II P = 12 - 1/9Q2= (72 + 1/27.216 – 0) - 48 5= (72 + 8 – 0) - 48 Q 0 6= 80 – 48= 32
20. 20. 6 P P = 12 - 1/9Q2Luas II = 6X8 - ∫(5 + 1/12Q2)dQ 2. 0 12 6= 48 – ( 5Q + 1/12.1/3Q3) ] 0 LUAS I= 48 – (5.6 + 1/36.63 – (5.0 + 8 1/36.03) LUAS II P = 5 + 1/12Q2= 48 – (30 + 1/36.216 – 0) 5= 48 - (30 + 6 - 0) Q 0 6= 48 – 36= 12
21. 21. 1.Fungsi pendapatan dari suatu pabrik diberikan sebagai berikut : R = 6 + 350Q – 2Q2Fungsi produksinya : Q = 3LJika jumlah tenaga kerja yang ada 25 orang,berapakah MPRL dan jelaskan artinya .Jawab :R = 6 + 350Q - 2Q² Q = 3LdR = 350 – 4Q dQ = 3dQ dL MPRL = dR = dR . dQ dL dQ dL = (350 – 4Q).3 L = 25 Q =3L = 75 dR = (350 – 300).3 = 175 dLArtinya: Untuk setiap penambahan Tenaga Kerja sebanyak 25 orang akan menyebabkan penambahan pendapatan sebanyak 175 ,dan sebaliknya