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KALKULUS 2

29. 22 au du =

2

1u

22 au 2

1a

2ln u +

22 au + c

Catatan : Dalam menyelesaikan soal integral diusahakan merubahnya menjadi salah

satu bentuk rumus di atas. Metoda ini disebut metoda substitusi

Contoh Soal

1. 2

1x3 dx =

2

1 x3 dx =

2

1.4

1x4 + c =

8

1x4 + c

2. 5

2

x

dx = 2x-5 dx = -2

1x

-4+ c = -

42

1

x

+ c

3. (2 + x) x . dx

= 2 x + x x . dx

= 2x1/2 + x3/2. dx

=

23

2x

3/2+

25

1x

5/2+ c

=3

4 .3x +

5

2 5x + c

=3

4 . x x +5

2x2 x + c

4. 2x21 x dx metoda substitusi

Misalnya u = 1 + x2

du = 2x dx

I = 2x u x

du

2

= u du

=

23

1u

3/2+ c

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KALKULUS 2

=3

2(1 + x

2)3/2

+ c

=

3

2 3)21( x + c

5. 3

62

3

1

yy

y

dy

Misal u = 3y2

+ 6y

du = 6y + 6 dy

= 6 (y + 1) dy

I = 3

1

u

y

)1(6 y

du

=6

1 u-1/3 du

=6

1.

32

1u

2/3+ c

=4

1. u

2/3+ c

=

4

1.

3 2)623( yy + c

6.

54

3

x

xdx

Misal u = x4

+ 5

du = 4x3

dx

I = u

x3

.3

4x

du

=4

1 udu =

4

1 ln u + c . =4

1 ln x4 + 5 + c

atau = ln c4

54 x

7.

1xe

dx

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KALKULUS 2

= x

e

xe

1

dx

Misal u = 1 + e-x

du = -e-x

dx

I = u

xe

.x

e

du

= u

du= ln u + ln c = In

xe

c

1

8. x .125

2 xdx

u = 2x2

+ 1

du = 4x dx

I = x . 5u .x

du

4

=4

1 5u . du

=4

1.

5ln

5u

+ c

=5ln4

1252 x

+ c

9. cos5

2x = @

25 sin

5

2x + c

10. cos3 x . sin x dx

u = cos x

du = -sin x dx

I = u 3 . sin x .x

du

sin

= u3 du

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KALKULUS 2

= 4

1u

4+ c

=

4

1cos4 x + c

11.

12

2

x

xdx

=

12

112

x

xdx

= 1 1

12 x

dx

= x arc tg x + c rumus 21

12.

z

zctg

2sin

dz

=

zz

z

2sin

1.

sin

cosdz

=

z

z

3sin

cosdz

= sin -3 z d (sin z)

= 2

1sin

-2z + c

= 2

1cosec2 z + c

13. 4

1 x

dxx

Misal u = x2

du = 2x dx

I = 2

1 u

x

.x

du

2

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KALKULUS 2

=2

1

21 u

du

=

2

1arc sin u + c

=2

1arc sin x2 + c

14. 1

52

3

x

xxdx

= 3x + 2 1

2

xdx

=23 x2 + 2x 2 ln x + 1 + c

= x + 1 3x2+ 5x 3x + 2

3x2 + 3x

2x

2x + 2

2

15.

xe

xe

21

5

dx

misal u = ex

du= ex

dx

I = 2

1

5

u

u

.u

du

= 5 2

1 u

du

= 5 arc sin u + c

= 5 arc sin e x + c

16. 26 xx

dx

x2

6x + 9 = (x 3)2

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KALKULUS 2

=

)269(9 xx

dx

= 2)3(9 x

dx

= 2)3(9

)3(

x

xd

= arc sin3

3x+ c

17.

842

1

xx

xdx

= 2

1

842

22

xx

x

dx

=2

1

842

242

xx

xdx

=2

1

842

42

xx

x

842

1

xxdx

=2

1lnx2 + 4x + 8

42)2(

1

xd (x + 2)

=2

1lnx2 + 4x + 8

2

1arc tg

2

2x+ c

18. 223 xx dx

=

2

)1(4

x dx rumus 27

=2

1(x+1)

223 xx + 2 . arc sin

2

1x+ c

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KALKULUS 2

SOAL-SOAL

1. xx 12

dx

2. 3 2)52( t dt

3. y83 dy

4.

x

x

2

2)21(dx

5.

3 222

88

zz

z

dz

6. z

tgz

cosdz

7.

1

22

x

xxdx

8.

2

22

3

x

xxdx

9. xe

xe

21

25dx

10.

)14(2sin1

)14(sin

t

tdt

11. 4916 y

ydy

12. ytg

y

229

22secdy

13. 522

1

xxdx