Antiderivatives for senior high school

1. Antiderivatives
5.1

2. Discovery of Power Rule for
Antiderivatives
If f ‘ (x) =
Then f(x) =
If f ‘ (x) =
Then f(x) =
If f ‘ (x) =
Then f(x) =
If f ‘ (x) =
Then f(x) =
3
2
9
4 2
3


 x
x
x
x
x
x
x 3
3 2
3
4



2
3 
 x
x
x
x
x 2
2
3
3
2 2
2
3


5
4
3
2
3 2


 x
x
x
x
x
x
x 5
2
2
3
5
3 2
2
3
5


 
5
3
4 2


 x
x
x
x
x 5
2
3
3
4 2
3




3. 1
of
tive
Antideriva
1



n
x
x
n
n

4. Differentiation
Integration
The process of finding
a derivative The process of finding
the antiderivative
Symbols:
Symbols:
)
(
'
,
'
, x
f
y
dx
dy
 dx
x
f )
(
Integral
Integrand
Tells us the
variable of
integration

5.  dx
x
f )
( is the indefinite integral of f(x) with respect to x.
Each function has more than one antiderivative (actually infinitely many)
2
3
2
3
2
3
2
3
3
58
3
4
3
6
3
x
x
x
x
x
x
x
x







Derivative of:
The
antiderivatives
vary by a
constant!

6. General Solution for an Indefinite Integral
C
x
F
dx
x
f 

 )
(
)
(
Where c is a constant
You will lose points if
you forget dx or + C!!!

7. Basic Integration Formulas
C
n
x
dx
x
n
n




 1
1
C
kx
dx
k 


C
a
ax
dx
ax 



cos
)
sin(
C
a
ax
dx
ax 


sin
)
cos(
C
a
ax
dx
ax 


tan
)
(
sec2
C
a
ax
dx
ax 



cot
)
(
csc2
C
a
ax
dx
ax
ax 


sec
)
tan
(sec
C
a
ax
dx
ax
ax 



csc
)
cot
(csc

8. Find:
 dx
x5
 dx
x
1
 xdx
2
sin
 dx
x
2
cos
C
x


6
6
C
x
C
x
dx
x 



 

2
2 2
1
2
1
C
x



2
2
cos
C
x
C
x




2
sin
2
2
1
2
sin
You can always check
your answer by
differentiating!

9. Basic Integration Rules

  dx
x
f
k
dx
x
kf )
(
)
(
  

 

 dx
x
g
dx
x
f
dx
x
g
x
f )
(
)
(
)
(
)
(

10. Evaluate:
 
 

 dx
x
x
x 4
6
5 2
3
 

 


 dx
dx
x
dx
x
dx
x 4
6
5 2
3
 

 


 dx
dx
x
dx
x
dx
x 4
6
5 2
3
C
x
C
x
C
x
C
x































 4
2
6
3
4
5
2
3
4
C
x
x
x
x




 4
3
3
4
5 2
3
4
C represents
any constant

11. Evaluate:
 dx
x
5 3

 5
3
x C
x


5
8
5
8
C
x 
 5
8
8
5

12. Evaluate:
 
  dx
x
x cos
3
sin
4
 

 xdx
xdx cos
3
sin
4
C
x
C
x 



 sin
3
cos
4
C
x
x 


 sin
3
cos
4

13. Evaluate:
 
  dx
x
2
2
3
 
 

 dx
x
x 4
2
6
9
C
x
x
x 



5
2
9
5
3
C
x
x
x



 9
2
5
3
5

14. Evaluate:
 dx
x
x
2
cos
sin
 
 dx
x
x
x cos
sin
cos
1
C
x
 sec
 
 xdx
x tan
sec

15. Particular Solutions
2
1
)
(
'
x
x
f  and F(1) = 0
dx
x
x
F 
 2
1
)
(
dx
x
x
F 

 2
)
(
C
x
x
F 



1
)
(
1
C
x
x
F 


1
)
(
0
1
1
)
1
( 


 C
F
1

C
1
1
)
( 


x
x
F