SUB-TOPIC
1. The anti derivative
2. Indefinite integrals
3. Define Integrals
4 Application of integration
THE ANTI-DERIVATIVE
. Is the reverse of differentiation.
-In differentiation we start with function to find the derivative
-For anti derivative we start with derivative to find the function
Consider the table below
| FUNCTION | DERIVATIVE | ANTI-DERIVATIVE (INTEGRATION) |
| y=x2 | y1=2x |  =x2+c |
| y=x3 | y1=3x2 |  =x3 +c |
| y=4 | y1=0 |  =c |
 |
y1=xn |  = xn+1+c |
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Integral notation
If y is the function of x, then 
is known as integration of y with respect to x
The integral sign cannot divorced with dx if we are integrating with respect to x.
Generally
If xn is integrated with respect to xn then
=
xn+1 + c
Examples
Find
a) 
Solution
=
x1+1 + c
=
+c
b) 
Solution
=
x2+1 + c
=
+ c
c) 
dv
Solution
dv = 
=
v2+1 + 
v1+1 + v + c
= 
+ 
+ v + c
EXERCISE
Integrate the following
1. 
2. 
3. 
4. 
Solution
1. 
= 
+ 7 
– 3x + c
= 
+ 7 
– 3x + c
2. 
dx
= 
+ 
7
+ c
= 
+ 
+ c
= 
-7
+ c
2. INDEFINITE INTEGRALS
Is an integral which does not have limits at the ends of the integral sign.
An arbitrary constant must be shown
e.g. 
,
, 
e.t.c
Example
Integrate the following with respect to X
1. 5
-7x+8
2. 2
– 
3. 4
Solution
1. 1.
= 
– 
+ 8x + c
= 
– 
+ 8x + c
2. 2
–
– 
+ 
+ c
+ 
+ c
Or
( 
)3 + 
+ c
3. 
dx
Let u=
u2=3x+1
2udu=3dx
dx = 
du
dx = 
= 
du
= 
du
= 
x 
+ c
But u =
EXERCISE
Integrate the following
1. 
– 5
+ 12) dx
2. 
3. 
4. 