The document discusses differentiation and integration in calculus. Differentiation is used to find the derivative or slope of a tangent line to a curve at a point, while integration is the inverse process of differentiation used to find the anti-derivative or area under a curve. Indefinite integration provides a family of curves that all have the same tangent slopes at each point rather than a unique curve.

1. He formulated the theory of calculus

2. DIFFERENTIATION
• TO FIND THE DERIVATIVE OF A
FUNCTION.
• DERIVATIVE IS THE SLOPE OF THE
TANGENT TO A CURVE AT A
PARTICULAR POINT.

3. INTEGRATION
• INTEGRATION IS AN INVERSE PROCESS
OF DIFFERENTIATION.
• INTEGRATION IS TO FIND THE ANTI-
DERIVATIVE OR INTEGRAL.
• IT IS THE PROCESS OF FINDING THE
FUNCTION WHEN SLOPE OF THE
TANGENT TO A CURVE AT A
PARTICULAR POINT IS GIVEN.

4. INDEFINITE INTEGRAL
df( x)
F ( x)
dx
F ( x) dx f ( x) c

5. INDEFINITE INTEGRAL
2
d( x )
2( x )
dx
2
2 x dx x c

6. INDEFINITE INTEGRAL
2
2 x dx x c
2 x dx x 2 0 when c 0
2
2 x dx x 1 when c 1
2 x dx x 2 1 whenc -1
WE GET A FAMILY OF CURVES.

7. GEOMETRICAL INTERPRETATION OF
INDEFINITE INTEGRAL
2 y = x +2
AT A, B, C, D, E…. THE SLOPE OF TANGENT IS SAME .
y = x2+1
y = x2
y=2x
D
y = x2 -1
B
y = x2-2 A
0
C
THE CURVES ARE DIFFERENT BUT THEY BELONG TO
SAME FAMILY.
E

8. COMPARISON BETWEEN DIFFERENTIATION
AND INTEGRATION
1. ALL THE FUNCTIONS ARE 1. ALL THE FUNCTIONS ARE
NOT DIFFERENTIABLE. NOT INTEGRABLE.
2. DERIVATIVE OF A FUNCTION 2. INTEGRAL OF A FUNCTION IS
IS UNIQUE. NOT UNIQUE.
3. WHEN POLYNOMIAL IS 3. WHEN POLYNOMIAL IS
DIFFERENTIATED WE GET A INTEGRATED WE GET A
POLYNOMIAL OF DEGREE 1 POLYNOMIAL OF DEGREE 1
LESS THAN IT. MORE THAN IT.
4. IT IS USED TO FIND 4. IT IS USED TO FIND
VELOCITY WHEN VELOCITY WHEN
DISPLACEMENT IS GIVEN ACCLERATION IS GIVEN