BAB 7 METODE INTEGRASI



BAB 7 METODE INTEGRASI

Rumus-rumus dasar Integral :

1. ( d/dx f(x)dx = f(x) + c

2. ( (U+V)dx = ( Udx + ( Vdx , U dan V fgs dari x

3. ( (Udx = ( ( Udx , ( konstanta, U fgs dari x

4. ( Undu = (Un+1/n+1) + c , n ( -1

5. ( du/u = ln (u(+c

6. ( audu = (au/ln a) + c , a>0, a ( 1

7. ( eudu = eu + c

8. ( sin u du = - cos u + c

9. ( cos u du = sin u + c

10. ( tg u du = ln (sec u ( + c

11. ( ctg u du = ln (sin u ( + c

12. ( sec u du = ln (sec u + tg u (+ c

13. ( cosec u du = ln (cosec u – ctg u(+ c

14. ( sec2 u du = tg u + c

15. ( cosec2 u du = -ctg u + c

16. ( sec u tg u du = sec u + c

17. ( cosec u ctg u du = - cosec u + c

Contoh :

1. ( x4dx = (x5/5) + c

2. ( 2xdx = 2( xdx = 2(x2/2) = x2 + c

3. ( 2x2+3x+1dx = 2( x2dx + 3( xdx + ( dx = 2/3x3 + 3/2x2 + x + c

4. ( (1-x)(xdx = ( (xdx - ( x(xdx = ( x1/2dx - ( x3/2dx

= x3/2/3/2 - x5/2/5/2 = 2/3x3/2 – 2/5x5/2 + c

5. ( 1/x4dx = ( x-4dx = x-3/-3 = (-1/3x3) + c

6. ( (2x3+3x2+xdx)/x = ( 2x2+3x+1dx = 2/3x3+3/2x2+x+c

7. ( (5x4-4x3+3x2dx)/x2 = ( 5x2-4x+3dx = (5/3)x3-(4/2)x2+3dx

= (5/3)x3-2x2+3x+c

8. ( 2x-(1/x3)dx = ( 2x - ( x-3dx = (2/2)x2-(x-2/-2)dx = x2+1/2x-2+c

9. ( (2-(x)xdx = ( 2xdx - ( x(xdx = (2/2)x2dx - ( x3/2dx

= x2-2/5x5/2+c

10. ( (3x2+4x-5dx)/x = ( 3x+4-(5/x)dx = ( 3x+4-5x-1dx

= 3/2x2+4x-5ln(x(+c

INTEGRAL DGN METODE SUBTITUSI

1. ( (x3+2)3.3x2dx ( U = x3+2

du/dx = 3x2 , dx = du/3x2

= ( U3.3x2.du/3x2

= ( U3du = U4/4 = {(x3+2)4/4} + c

2. ( 6x2 dx ( U = x3-3

(x3-3) du/dx = 3x2 , dx = du/3x2

= ( 6x2 . du = 2 ( du = 2 ln(u(= 2ln(x3-3(+c

U 3x2 U

3. ( 5x.(1-3x2dx ( U = 1-3x2

du/dx = -6x , dx = du/-6x

= ( 5x.(U.du = ( -5(U.du = ( -5.U1/2.du

-6x 6 6

= -5/6.2/3.U3/2 = -5/9U3/2 = -5/9(1-3x2)(1-3x2

4. ( cos3xdx ( U = 3x

du/dx = 3, dx = du/3

= ( cos U.du/3 = 1/3( cos udu = 1/3sin udu

= 1/3sin3x + c

5. ( e3xdx ( U = 3x

du/dx = 3 , dx = du/3

= ( eu.du/3 = 1/3( eudu = 1/3e3x + c

6. ( 4xsin(x2+4)dx ( U = x2+4

du/dx = 2x , dx = du/2x

= ( 4x.sin u.du/2x = 2( sinudu = -2cos u

= -2cos(x2+4) + c

7. ( dx ( U = x + 3

(x+3) du = dx

= ( du/u = ln(u(= ln (x+3(+c

8. ( x dx ( U = x2-1

(x2-1) du/dx = 2x , dx = du/2x

= ( x/u.du/2x = ½ ln(u(+ ln (c(

= ½ ln (x2-1(+ ln (c(

9. ( cos 1/2xdx ( U = 1/2x

du/dx = ½ , dx = du/1/2 ( dx = 2du

= ( cos u.2du = 2 ( cos udu = 2 sin udu

= 2 sin 1/2x + c

INTEGRAL PARSIAL

Rumus : ( Udv = UV - ( Vdu

Mengandung : ~ Polinom dg trigonometri (xn.sinx)

~ Polinom dg eksponensial (xn.ex)

~ Eksponensial dg trigonometri (ex.sinx)

~ logaritma atau Ln

1. ( xcos xdx ( u = x , du = dx

dv = cos xdx , v = sin x

= xsinx - ( sin xdx

= xsinx + cosx + c

2. ( xsinxdx ( u = x , du/dx = 1 , du = dx

dv = sinxdx , v = -cosx

= -xcosx - ( cos xdx

= -xcosx + sinx + c

3. ( xexdx ( u = x , du/dx = 1 , du = dx

dv = exdx , v = ex

= xex - ( exdx

= xex – ex + c

4. ( xlnxdx ( u = lnx , du/dx = 1/x , du = (1/x)dx

dv = xdx , v = 1/2x2

= lnx.1/2x2 - ( 1/2x2.1/xdx

= lnx.1/2x2 - ½ ( xdx

= lnx.1/2x2 - ½(1/2x2)

= lnx.1/2x2 - 1/4x2 + c

5. ( x2e2xdx ( u = x2 , du/dx = 2x , du = 2xdx

dv = e2xdx , v = (½)e2x

= x2.1/2e2x - ( ½.e2x.2xdx

= x2.1/2e2x - ( x.e2xdx …..pasrsialkn lg

= x2.1/2e2x - { x.½.e2x - ( ½.e2xdx }

= x2.1/2e2x - { x.½.e2x – ¼ e2x }

= x2.1/2e2x + x.½.e2x – ¼ e2x + c

6. ( x2lnxdx ( u = ln x , du/dx = 1/x , du = (1/x)dx

dv = x2dx , v = (1/3)x3

= lnx.1/3x3 - ( 1/3x3.1/xdx

= lnx.1/3x3 - 1/3 ( x2dx

= lnx.1/3x3 - 1/3 (1/3x3)

= lnx.1/3x3 - 1/9x3 + c

INTEGRAL FUNGSI TRIGONOMETRI

Rumus reduksi :

a. ( cosn xdx = 1/n.cosn-1x.sinx + n-1/n ( cos n-2 xdx

b. ( sinn xdx = -1/n.sinn-1x.cosx + n-1/n ( sin n-2 xdx

c. ( dx/cosnx = (secnxdx = 1/n-1(sinx/cosn-1x)+n-2/n-1(secn-2xdx

d. ( dx/sinnx = (cosecnxdx =-1/n-1(cosx/cosn-1x)+n-2/n-1(cosecn-2xdx

Rumus lainnya :

1. sin2x + cos2x = 1

2. 1 + tg2x = sec2x

3. 1 + ctg2x = cosec2x

4. sin2x = ½ (1 – cos2x)

5. cos2x = ½ (1 + cos2x)

6. sin x cos x = ½ sin2x

7. sin x cos y = ½ {sin(x-y) + sin(x+y)}

8. sin x sin y = ½ {cos(x-y) - cos(x+y)}

9. cos x cosy = ½ {cos(x-y) + cos(x+y)}

10. 1 – cos x = 2 sin2 ½ x

11. 1 + cos x = 2 cos2 ½ x

12. 1 ( sin x = 1 ( cos (1/2)n –x

Rumus reduksi lainnya :

( sinmx cosnxdx = sinm+1x.cosm-1x + n-1 ( sinmx.cosm-1xdx

m+n m+n

Contoh :

1. ( cos5xdx = 1/5cos4x.sinx + 4/5(cos3xdx …….rms (a)

= 1/5cos4x.sinx + 4/5{1/3cos2x.sinx+2/3(cosxdx}

= 1/5cos4x.sinx + 4/15cos2x.sinx+8/15sinx+c

2. ( cos3xdx = 1/3cos2xsinx + 2/3(cosxdx ………rms (a)

= 1/3cos2xsinx + 2/3sinx + c

3. ( sin2xdx = -1/2sinxcosx + ½ ( sin0xdx …….rms (b)

= -1/2sinxcosx + ½ ( 1dx

= -1/2sinxcosx + ½ x + c (atau)

= -1/2(1/2sin2x)+ ½ x

=-1/4sin2x + ½ x + c

= ( ½ (1-cos2x)dx …. Rms 4

= ( ½ dx – ½ ( cos2xdx … (1/a sinax)

= ½ x – ¼ sin2x + c

= ½ x – ¼ (2sinxcosx)

= ½ x – ½ sinx cosx + c

= -½ ( ½ sin2x) + ½ x … rms 6

= -¼ sin2x + 1/2x + c

4. ( sin3x cosxdx = ( ½ (sin2x + sin4x)dx … rms 7

= ½ ( sin2xdx + ½ ( sin4xdx

= ½ (-1/2cos2x) + ½ (-1/4cos4x)

= -1/4cos2x – 1/8cos4x + c

5. ( sin5x sin3xdx = ( ½ (cos2x – cos8x)dx … rms 8

= ½ ( cos2xdx – ½ ( cos8xdx

= ½ (½ sin2x) – ½ (1/8 sin 8x)

= ¼ sin2x – 1/16 sin8x + c

6. ( cos4x cos2xdx = ( ½ (cos2x + cos 6x)dx … rms 9

= ½ ( cos2xdx + ½ ( cos6xdx

= ½ (½ sin2x) + ½ (1/6sin6x)

= ¼ sin2x + 1/12 sin6x + c

7. ( sin6xdx = -1/6 sin5x cosx + 5/6 ( sin4xdx

= -1/6 sin5x cosx + 5/6 {-1/4sin3xcosx + 3/4(sin2xdx}

= -1/6 sin5x cosx + 5/6 {-1/4sin3xcosx + 3/4(1/2(1-cos2xdx}

= -1/6 sin5x cosx + 5/6 {-1/4sin3xcosx + 3/4(1/2dx-1/2(cos2xdx

= -1/6 sin5x cosx + 5/6 {-1/4sin3xcosx + ¾(1/2x-1/4sin2x)}

= -1/6 sin5x cosx + 5/6 (-1/4sin3xcosx + 3/8x – 3/16sin2x}

= -1/6 sin5x cosx – 5/24sin3xcosx + 15/48x – 15/96sin2x + c

INTEGRAL FUNGSI RASIONAL

FUNGSI RASIONAL SEJATI :

(1) Faktor linear yg berbeda

Jika g(x) = (a1x+b1)(a2x+b2)…(anx+bn)

Maka f(x) = A1 + A2 + …. An

A1x+b1 a2x+b2 Anx+bn

Contoh :

1. F(x) = ( x+1 dx = x+1 = A + B

X2-4x-12 (x-6)(x+2) x-6 x+2

= A(x+2) + B(x-6) = (A+B)x + (2A-6B)

(x-6)(x+2) x2-4x-12

(A+B)x = x 2A-6B = 1

A+B = 1 2(1-B)-6B = 1

A = 1-B 2-2B-6B = 1

A =1-1/8 -8B = -1

A = 7/8 B = 1/8

F(x) = ( x+1 dx = 7dx + dx = 7/8ln(x-6(+1/8ln(x+2(+c

x2-4x-12 8(x-6) 8(x+2)

2. F(x) = ( x – 4 dx = x – 4 = A - B

x2+x-12 (x+4)(x-3) x+4 x-3

= A(x-3)-B(x+4) = (A-B)x – (-3A-4B)

(x+4)(x-3) x2+x-12

(A-B)x = x 3A+4B = -4

A-B = 1 3(1+B)+4B = -4

A = 1+B 3+3B+4B = -4

7B = -4+3

A = 1 + B B = -1/7

= 1 – 1/7

= 8/7

F(x) = ( x – 4 dx= ( 8/7 dx + (1/7 dx = 8/7ln(x+4(+1/7ln(x-3(+c

x2+x-12 (x+4) (x-3)

FAKTOR LINIER BERULANG :

Jika g(x) = (ax+b)m ( berulang sebanyak m kali

Maka F(x) = A + B + ……. + Am

(a1x+b1) (a2x+b2)2 (amx+bm)m

Contoh :

1. ( x dx = A + B = A(x-3) + B

(x-3)2 (x-3) (x-3)2 (x-3)2

Ax-3A+B = x -3A+B = 0

Ax = x -3(1)+B = 0

A = 1 B = 3

( x dx = ( 1dx + ( 5 dx ( u = x-1 maka = ( 5 du

(x-1)2 (x-1) (x-1)2 du = dx u2

= 5 ( u-2du

= 5u-1 = -5 = -5

-1 u x-1

= ln ( x-1 ( - 5 + c

x-1

INTEGRAL TERTENTU

a(b f(x)dx = f(x) (ba = f(b) – f(a)

Contoh :

1. 1(2 x2+2xdx = x3 + 2x2 = 1x3 + x2 (21

3 2 3

= { 1/3(2)3 + (2)2 } – { 1/3(1)3 + (1)2 }

= { 8/3 + 4 } – { 1/3 + 1 }

= 20/3 – 4/3

= 16/3

2. 4(8 x dx ( u = x2-15 maka 4(8 x . du = ½ 4(8 u-1/2du

(x2-15 du/dx = 2x u1/2 2x = ½ u1/2

dx = du/2x ½

= (x2-15 (84

= ((64-15)-((16-15)

= 7-1 = 6

3. 1(8 ( 1+3x dx = 1(8 ( 1 + 3x)1/2 dx ( u = 1+3x

= 1(8 u1/2 . du/3 du/dx = 3

= 1/3 1(8 u1/2 du dx = du/3

= 1/3 u3/2

3/2

= 2/9 u3/2

= 2/9 u 2/2.u1/2

= 2/9 (1+3x)((1+3x) (81

= { 2/9(1+3.8)(1+3.8 } – { 2/9(1+3)(1+3 }

= { 2/9.25.5 } – { 2/9.4.2 }

= 250/9 – 16/9

= 234/9

= 26

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