Introduction to integral calculus. This slideshow deals with concept of integration. A complete explanation is provided that how integration can be written as summation. Area under the graph can be calculated through integration.

2. INTRODUCTION TO
INTEGRAL CALCULUS
by - Mohammed Waris Senan

3. Integral Calculus
Mohammed Waris Senan
3
y
x
O
f (a)
f (b)
a
b
P
Q
A B
y = f (x)
x
O
f (a)
f (b)
a
b
Δx
P
Q
A B

4. Integral Calculus
Mohammed Waris Senan
4
y = f (x)
x
O
f (a)
f (b)
a
b
Δx
N
a
b
x



x
x
N
a
f
x
x
a
f
x
x
a
f
x
a
f
I'
















]
)
1
(
[
.....
...
)
2
(
)
(
)
(
x
N
a
x
a
x
a
a
x
Where
x
x
f
I'
as
written
be
may
This
i
N
i
i








 


)
1
(
,.......,
2
,
,
,
)
(
1
P
Q
A B
This area differs slightly from the area PABQ. This difference is
the sum of the small triangles formed just under the curve.

5. Integral Calculus
Mohammed Waris Senan
5
y = f (x)
x
O
f (a)
f (b)
a
b
Δx
P
Q
A B
 As we increase the number of intervals N, the vertices of the
bars touch the curve PQ at more points and the total area of
the small triangles decreases.
 Now as N → ∞, Δx → 0 and the vertices of the bars touch
the curve at infinite number of points and the total area of
the triangles tends to zero.
Thus, we may write, area of PABQ under such limit as:
 








N
i
b
a
i
N
or
x
dx
x
f
x
x
f
I
1
0
)
(
)
(
lim

6. Integral Calculus
Mohammed Waris Senan
6
y = f (x)
x
O
f (a)
f (b)
a
b
Δx
P
Q
A B
Here, f(x) = x
 


 












N
i
b
a
i
N
or
x
N
i
i
N
or
x
xdx
x
x
x
x
f
I
1
0
1
0
lim
)
(
lim
x
x
N
a
x
x
a
x
x
a
x
a
I'
















]
)
1
(
[
.....
...
)
2
(
)
(
x
x
N
a
a
N
I' 





 }]
)
1
(
{
[
2

7. Integral Calculus
Mohammed Waris Senan
7
x
x
N
a
a
N
I' 





 }]
)
1
(
{
[
2
]
2
[
2
x
x
N
a
x
N
I' 






]
2
[
2
x
a
b
a
a
b
I' 





 








x
a
b
N

]
[
2
x
b
a
a
b
I' 





Thus, area PABQ is
]
[
2
lim
0
x
b
a
a
b
I
x








 



)
(
2
b
a
a
b
I 





 


 
2
1 2
2
a
b
I 


 
2
1
, 2
2
a
b
xdx
write
can
we
So
b
a




8. Integral Calculus
Mohammed Waris Senan
8
)
(
)
(
)]
(
[
)
( a
F
b
F
x
F
dx
x
f b
a
b
a




Now, let the derivative of F(x) is f(x)
i.e. , Then
F(x) is called the indefinite integration or the
anti-derivative of f(x).
)
(
)
( x
F
dx
x
f 

x
x
x
dx
d
x
dx
d
example
For










2
2
1
)
(
2
1
2
1
,
2
2
)
(
2
1
2
1
2
1
2
1
,
2
2
2
2
2
a
b
a
b
x
xdx
Thus
b
a
b
a

























9. Some useful integration formulae
Mohammed Waris Senan
9
x
xdx

10. Some useful integration formulae
Mohammed Waris Senan
10
x
xdx

11. Some useful integration formulae
Mohammed Waris Senan
11
x
xdx
2

12. Some useful integration formulae
Mohammed Waris Senan
12
x
xdx
2

13. Some useful integration formulae
Mohammed Waris Senan
13
x
xdx
x

14. Some useful integration formulae
Mohammed Waris Senan
14
x
xdx
x cosec
cot
cosec 

 

15. Some useful integration formulae
Mohammed Waris Senan
15
1
1
1






n
and
n
x
dx
x
n
n

16. Some useful integration formulae
Mohammed Waris Senan
16
x
dx
x
ln
1



17. Some useful integration formulae
Mohammed Waris Senan
17
a
x
a
dx
a
x
1
2
2
tan
1
1 




18. Some useful integration formulae
Mohammed Waris Senan
18
a
x
dx
x
a
1
2
2
sin
1 




19. Some useful rules
Mohammed Waris Senan
19
constant
a
is
Where
)
(
)
(
c
dx
x
f
c
dx
x
cf 



20. Mohammed Waris Senan
20
c
cx
F
dx
cx
f
then
x
F
dx
x
f
Let
)
(
)
(
)
(
)
(




Some useful rules

21. Mohammed Waris Senan
21
 


  dx
x
g
dx
x
f
dx
x
g
x
f )
(
)
(
)]
(
)
(
[
Some useful rules

22. Mohammed Waris Senan
22