College Mathematics: Learning Worksheets Chapter 12
Name ________________________________ Date ______________ Class ____________
Goal: To find antiderivatives and indefinite integrals of functions using the formulas and
properties
In Problems 1–3, find each indefinite integral and check by differentiating.
1. 26
x
dx
2
26
yxdx
Check: 2
13
y
xC
dy x

Section 12-1 Antiderivatives and
Indefinite Integrals
Theorem 1 Antiderivatives
If the derivative’s of two functions are equal on an open interval (a, b), then the
functions differ by at most a constant. Symbolically, if F and G are differentiable functions
on the interval (a, b) and ‘( ) ‘( )Fx Gxfor all x in (a, b), then ( ) ( )Fx Gx kfor some
constant k.
Formulas and Properties of Indefinite Integrals
For C and k both a constant
1
n
nx
2. xx
edx e C
5. [ () ()] () ()
f
x gx dx f xdx gxdx 

2. 12
9
dx
12
32
y
9
y
xdx
Check: 32
12
6
y
xC

3. 7x
edx
7
y
x
y
edx
Check:
7
x
y
eC

4. Find all the antiderivatives for 1
73.
dy z
dz

1
73
dy z
dz

College Mathematics: Learning Worksheets Chapter 12
In Problems 5–8, find each indefinite integral.
5. 53 2
(76)
x
xx dx
532 875
(76) (76)
x
x
xx dxxxxdx
  

6. 5
2
3
x
dx
x



52
72
52
2
1
72
7
33
3
1
32
7
x
dx x x dx
x
xx
C
xC
x






 
7. 4
92xdx
x



92 9 2
xdx dx
 
College Mathematics: Learning Worksheets Chapter 12
346
8.
56
6
53
x
xxe
dx
x



9. 32
‘( ) 4 12 5;Rx x x  (2) 50R
First, find the indefinite integral of the function.
32
R
() (4 12 5)
R
xxxdx

College Mathematics: Learning Worksheets Chapter 12
347
10. 548;
t
dy et
dt  (0)8y
First, find the indefinite integral of the function.
(5 4 8)
t
dy e t dt

Using the condition given, find the specific value of C.
2
528
t
ye t tC

College Mathematics: Learning Worksheets Chapter 12
348
11.
3
3
65
;
dD x x
dx x
(10) 50D
First, find the indefinite integral of the function.
3
3
65
xx
dD dx
x




Using the condition given, find the specific value of C.
5
() 6
Dx x C
x

College Mathematics: Learning Worksheets Chapter 12
12. 12
‘( ) 6 7 ;hx x x

 (1) 3h
First, find the indefinite integral of the function.
12
‘( ) 6 7
hx x x


College Mathematics: Learning Worksheets Chapter 12
350
College Mathematics: Learning Worksheets Chapter 12
Name ________________________________ Date ______________ Class ____________
Goal: To find the indefinite integrals using general indefinite integral formulas
Section 12-2 Inte
g
ration b
y
Substitution
Formulas: General Indefinite Integral Formulas
1
[()]
n
nfx
3. 1‘( ) ln ( )
f
xdx f x C
1
1
n
nu
n
5. uu
edu e C
u
Definition: Differentials
If ()yfxdefines a differentiable function, then
2. The differential dy of the dependent variable y is defined as the product of
Procedure: Integration by Substitution
1. Select a substitution that appears to simplify the integrand. In particular, try to
select u so the du is a factor in the integrand.
3. Evaluate the new integral if possible.
4. Express the antiderivative found in step 3 in terms of the original variable.
College Mathematics: Learning Worksheets Chapter 12
Check:
54
55
4
5
4(5 7 3)
5
4(5 7 3)
5
yxxC
yxx C


3.
3
4
34
6327
tdt
tt

Let
4
6327,ut ttherefore, 33
(24 32) 8(3 4) .du t dt du t dtRewrite the
original integral in terms of the variable u and solve.
3
4
18(3 4)
86327
y
t
y
dt
tt

Check:
4
4
4
3
ln 6 32 7
8
1(6 32 7)
8(6 32 7)
1(24 32)
tt
yC
dy d tt
dt dx
tt
dy t




354
4. 1.5x
edx
Let 1.5 ,ux therefore, 1.5 .du dx Rewrite the original integral in terms of the
variable u and solve.
1.5
1.5
1(1.5 )
1.5
x
x
ye dx
y
edx
 
Check:
1.5
2
3
x
ye C
 
5. 7
(7)
x
xdx
Let 7,ux therefore, .du dxIf we rewrite the original integral in terms of the
variable u, the substitution would not be complete. We also need 7.ux Now rewrite the
original integral in terms of u and solve.
7
7
(7)
(7)
yxx dx
yuudu


College Mathematics: Learning Worksheets Chapter 12
355
Check:
98
(7) 7(7)
98
xx
yC

 
6.
24
2(ln(3 ))xdx
x
Let 2
ln(3 ),uxtherefore, 2
62
.
3
x
du dx du dx
x
x
Rewrite the original integral in
terms of the variable u and solve.
24
2(ln(3 ))
y
x
y
dx
x
Check:
5
2
ln 3
5
x
yC

College Mathematics: Learning Worksheets Chapter 12
7.
15
6
1x
edx
x
Let 5,ux
 therefore, 6
65
5.
x
du x dx du dx
Rewrite the original integral in
terms of the variable u and solve.
15
6
1
y
x
y
edx
15
1
5
1
5
x
u
y
eC

Check:
15
1
x
ye C

357
8. 23 5 23 5
12 (2 7) 12 (2 7)
dy
x
xdyxxdx
dx 
Let 3
27,uxtherefore, 2
6.du x dxRewrite the original integral in terms of the
variable u and solve.
23 5
y
12 (2 7)
y
xx dx

Check:
36
1(2 7)
3
yx C

9. The indefinite integral can be found in more than one way. Given the integral,
22
2( 3) ,
x
xdx
first use the substitution method to find the indefinite integral and
then find it without using substitution.
Using substitution, let 23,uxtherefore, 2.du x dxRewrite the original integral
in terms of the variable u and solve.
22
2( 3)
y
xx dx

College Mathematics: Learning Worksheets Chapter 12
358
Expanding the final function yields the following:
23
222
y
1(3)
3
1(3)(3)(3)
3
yx C
y
xxx C


22 2 2
2 ( 3) 2 ( 3)( 3)
xx xx x
  
Now find the indefinite integral using the above function as the integrand.
53
y
(2 12 18 )
y
xxxdx
