D i b u a t O l e h P a k S u k a n i ; E m a i l : L i k e n y _ r b g y a h o o . c o m Page 1
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INTEGRAL
12.1. Integral Tak Tentu a. Integral Fungsi Aljabar
Rumus Dasar Integral tak tentu :
1.
C x
1 n
a dx
ax
1 n
n
n -1 2.
C ax
dx a
3.
C
x ln
dx x
dx x
1
-1
Contoh : 1.
dx
x x
x 7
4 6
2
2 3
=
C x
x x
x
7
2 2
2 1
2 3
4
2.
dx x
x
3 5
2 2
3
2
=
C x
x x
3
2 5
2 3
2 3
2 3
=
C x
x x
3
5 1
2 1
2 3
3.
dx x
x 3
2 5
4
4 1
3 1
=
C x
x x
3
2 4
5 5
3 4
4
4 5
3 4
=
C x
x x
3
2 4
3
4 5
3 4
4.
dx x
x
5 3
4 =
dx x
x
20 12
2
= 3x
3
+ 10x
2
+ C 5.
dx x
x 6
4 2
=
dx
x x
x 24
4 12
2
2
=
dx x
x 24
8 2
2
=
C x
x x
24
4 3
2
2 3
6.
dx x
2
5 =
dx x
x
25 10
2
=
C x
x x
25
5 3
1
2 3
7.
dx x
2
2 3
=
dx x
x 4
12 9
2
= 3x
3
+ 6x
2
+ 4x + C 8.
dx x
x x
3 5
2
3 2
=
dx x
x x
3 5
2
3 2
1
=
C x
x x
2 1
2 3
1 5
ln 2
=
C x
x x
2
2 3
5 ln
2
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9.
dx x
x 2
3
=
dx x
x
2
3 1
2 1
=
C x
x
3 4
2 3
3 4
2 2
3 1
=
C x
x
3 4
2 3
3 2
3 2
b. Integral Fungsi Trigonometri Rumus dasar integral :
1.
dx x
sin
= – cos x + C
2.
C
ax cos
a 1
dx ax
sin
;
C
b ax
cos a
1 dx
b ax
sin
3.
dx x
cos
= sin x + C 4.
C ax
sin a
1 dx
ax cos
;
C b
ax sin
a 1
dx b
ax cos
Contoh : 1.
xdx 3
sin 2
=
C x
3 cos
3 2
2.
xdx 5
cos 3
=
C x
5
sin 5
3
3.
dx
x 2
5 sin
4
=
C x
2 5
cos 2
4
= 2cos5 – 2x + C
4.
dx x
1 2
1 cos
2
=
C x
1 2
1 sin
2 1
2
=
C x
1 2
1 sin
4
5.
dx x
x 3
4 sin
3 2
cos 5
=
C x
x
3
4 cos
4 3
2 sin
2 5
=
C x
x
3 4
cos 4
3 2
sin 2
5
12.2. Integral Tertentu Integral Batas
Rumus Dasar : Fa
Fb Fx
fxdx
b a
b a
a = batas bawah b = batas atas
Contoh :
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1.
3 1
2
4 6
3
dx x
x
=
3 1
2 3
4 3
x x
x
= 3
3
– 1
3
+ 33
2
– 1
2
– 43 – 1 = 27
– 1 + 39 – 1 – 43 – 1 = 26 + 24
– 8 = 42
2.
4
3 4
2
dx x
x
=
4 2
12 4
6 2
dx x
x x
=
4 2
12 2
2
dx x
x
=
4 2
3
12 3
2
x x
x
=
3 2
4
3
– 0
3
+ 4
2
– 0
2
– 124 – 0 =
3 2
64 – 0 + 16 – 0 – 124 – 0
= 42
3 2
+ 16 – 48
= 10
3 2
3.
2 1
2
3 2
dx x
=
2 1
2
9 12
4
dx x
x
=
2 1
2 3
9 6
3 4
x x
x
=
3 4
2
3
– -1
3
– 62
2
– -1
2
+ 92 – -1
=
3 4
8 + 1 – 64 – 1 + 92 + 1
= 12 – 18 + 27
= 21 4.
4 1
3
dx x
=
4 1
2 1
3
dx x
=
4
1 2
3
2 3
3
x
=
4 1
2 3
2
x
=
4 1
3
2
x
= 1
4 2
3 3
=
1 2
2
3 2
= 22
3
– 1 = 28 – 1 = 14
D i b u a t O l e h P a k S u k a n i ; E m a i l : L i k e n y _ r b g y a h o o . c o m Page 4
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5.
3 1
3 2
6 2
dx x
x
=
3 1
3 2
6 2
dx x
x
=
3 1
2 1
2 6
1 2
x x
=
3 1
2
6 2
x x
=
1 1
3 1
6 1
1 3
1 2
2
=
1 9
1 6
1 3
1 2
=
9 8
6 3
2 2
=
3 16
3 4
= 3
12
= –4
6.
2
cos 2
xdx
=
2
sin 2
x
= 2sin 90
o
– sin 0
o
= 21 – 0
= 2 7.
2
2 sin
3
xdx
=
2
2 cos
2 3
x
= 2
3 cos 2180
o
– cos 290
o
= 2
3 cos 360
o
– cos 180
o
= 2
3 1 – -1
= –3
8.
3
sin 4
3 cos
2
dx x
x
=
3
cos 4
3 sin
3 2
x x
=
3 2
sin 360
o
– sin 30
o
+ 4cos 360
o
– cos 30
o
=
3 2
sin 180
o
– sin 0
o
+ 4cos 180
o
– cos 0
o
=
3 2
– 0 + 4–1 – 1 =
–8
D i b u a t O l e h P a k S u k a n i ; E m a i l : L i k e n y _ r b g y a h o o . c o m Page 5
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12.3. Pemakaian Integral a. Luas Daerah
