Chapter 7: Integration
Z n+1 Be careful!
n 1 (ax + b)
A (ax + b) = ¢ + c; n 6
= ¡1 To create f 0 (x) you can
a n+1
multiply and divide by
Z numbers. Not functions
f n+1 (x)
f 0 (x) ¢ f n (x) = + c; n 6
= ¡1 including x.
n+1
c
Example 1:
Éc É
Z
4
(a) (5x + 2) dx
Z
¡ ¢4
(b) 6x2 2x3 ¡ 7 dx
Yo by
Z
¡ ¢6
5
2
(c) (4x + 3) 2x + 3x ¡ 5 dx
my my
Z
5x2 2
3
452x
(d) p
x3 + 4
dx
55 Ic
dx E
ÉeÉ InexET
C
Z
age.EE
(e) cos x ¢ sin 5xdx
SEE c
siII c
Z ³ ´
¡ 2 ¢3 5
(f) 6x 3x ¡ 5 + (3x + 2) dx
2
6 13 513 dx 513 215dx
S
c
2444 c
, Z
B ex dx = ex + c
Z
ef (x) ¢ f 0 (x) dx = ef (x) + c
Example 2
Z
Yc
dx e
252
(a) e2x dx
(b)
Z
e3x+1 dx 538 dx 371C
Z ³
(c) 3 x4
4x e
´
dx ex
Ex
Z ³ ´ s
sdx
(d) 4 x5 +5
xe dx
3SEE.de fex
Z
t252Lx 2 ex44x7d
2
(e) (x + 2) ex +4x¡7 dx
44
1SC2XtY 844 74 12 74
Z
¡ ¢ 4
et
f dx
3 e
(f) 1 ¡ 4x3 ex ¡x dx i 4
Z
(g) cos xesin x dx esinx
Yy
Z n+1 Be careful!
n 1 (ax + b)
A (ax + b) = ¢ + c; n 6
= ¡1 To create f 0 (x) you can
a n+1
multiply and divide by
Z numbers. Not functions
f n+1 (x)
f 0 (x) ¢ f n (x) = + c; n 6
= ¡1 including x.
n+1
c
Example 1:
Éc É
Z
4
(a) (5x + 2) dx
Z
¡ ¢4
(b) 6x2 2x3 ¡ 7 dx
Yo by
Z
¡ ¢6
5
2
(c) (4x + 3) 2x + 3x ¡ 5 dx
my my
Z
5x2 2
3
452x
(d) p
x3 + 4
dx
55 Ic
dx E
ÉeÉ InexET
C
Z
age.EE
(e) cos x ¢ sin 5xdx
SEE c
siII c
Z ³ ´
¡ 2 ¢3 5
(f) 6x 3x ¡ 5 + (3x + 2) dx
2
6 13 513 dx 513 215dx
S
c
2444 c
, Z
B ex dx = ex + c
Z
ef (x) ¢ f 0 (x) dx = ef (x) + c
Example 2
Z
Yc
dx e
252
(a) e2x dx
(b)
Z
e3x+1 dx 538 dx 371C
Z ³
(c) 3 x4
4x e
´
dx ex
Ex
Z ³ ´ s
sdx
(d) 4 x5 +5
xe dx
3SEE.de fex
Z
t252Lx 2 ex44x7d
2
(e) (x + 2) ex +4x¡7 dx
44
1SC2XtY 844 74 12 74
Z
¡ ¢ 4
et
f dx
3 e
(f) 1 ¡ 4x3 ex ¡x dx i 4
Z
(g) cos xesin x dx esinx
Yy
