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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