PPT: Integral Calculus GATE Notes | EduRev

Engineering Mathematics

GATE : PPT: Integral Calculus GATE Notes | EduRev

 Page 1


Theorem 
If F is a particular antiderivative of f on an 
interval I, then every antiderivative of f on I
is given by
where c is an arbitrary constant, and all the 
antiderivatives of f on I can be obtained by 
assigning particular values for c. .
() F x c
Page 2


Theorem 
If F is a particular antiderivative of f on an 
interval I, then every antiderivative of f on I
is given by
where c is an arbitrary constant, and all the 
antiderivatives of f on I can be obtained by 
assigning particular values for c. .
() F x c
Notation
4 The symbol       denotes the operation of 
antidifferentiation, and we write 
where F’( x)=f ( x),  and c is an arbitrary constant. 
This is read “The indefinite integral of f(x) 
with respect to x is F(x) + c".
( ) ( ) f x dx F x c
Page 3


Theorem 
If F is a particular antiderivative of f on an 
interval I, then every antiderivative of f on I
is given by
where c is an arbitrary constant, and all the 
antiderivatives of f on I can be obtained by 
assigning particular values for c. .
() F x c
Notation
4 The symbol       denotes the operation of 
antidifferentiation, and we write 
where F’( x)=f ( x),  and c is an arbitrary constant. 
This is read “The indefinite integral of f(x) 
with respect to x is F(x) + c".
( ) ( ) f x dx F x c
In this notation, 
is the integral sign;
f(x) is the integrand; 
dx is the differential of x which denotes 
the variable  of integration; and 
c is called the constant of integration.
4 If the antiderivative of the function on interval 
I exists, we say that the function is integrable
over the interval I. 
( ) ( ) f x dx F x c
Page 4


Theorem 
If F is a particular antiderivative of f on an 
interval I, then every antiderivative of f on I
is given by
where c is an arbitrary constant, and all the 
antiderivatives of f on I can be obtained by 
assigning particular values for c. .
() F x c
Notation
4 The symbol       denotes the operation of 
antidifferentiation, and we write 
where F’( x)=f ( x),  and c is an arbitrary constant. 
This is read “The indefinite integral of f(x) 
with respect to x is F(x) + c".
( ) ( ) f x dx F x c
In this notation, 
is the integral sign;
f(x) is the integrand; 
dx is the differential of x which denotes 
the variable  of integration; and 
c is called the constant of integration.
4 If the antiderivative of the function on interval 
I exists, we say that the function is integrable
over the interval I. 
( ) ( ) f x dx F x c Integration Rules
1. Constant Rule. If k is any real number, then 
the indefinite integral of k with respect to x is  
2. Coefficient Rule. Given any real number 
coefficient a and integrable function f, 
kdx kx C
( ) ( ) af x dx a f x dx
Page 5


Theorem 
If F is a particular antiderivative of f on an 
interval I, then every antiderivative of f on I
is given by
where c is an arbitrary constant, and all the 
antiderivatives of f on I can be obtained by 
assigning particular values for c. .
() F x c
Notation
4 The symbol       denotes the operation of 
antidifferentiation, and we write 
where F’( x)=f ( x),  and c is an arbitrary constant. 
This is read “The indefinite integral of f(x) 
with respect to x is F(x) + c".
( ) ( ) f x dx F x c
In this notation, 
is the integral sign;
f(x) is the integrand; 
dx is the differential of x which denotes 
the variable  of integration; and 
c is called the constant of integration.
4 If the antiderivative of the function on interval 
I exists, we say that the function is integrable
over the interval I. 
( ) ( ) f x dx F x c Integration Rules
1. Constant Rule. If k is any real number, then 
the indefinite integral of k with respect to x is  
2. Coefficient Rule. Given any real number 
coefficient a and integrable function f, 
kdx kx C
( ) ( ) af x dx a f x dx
Integration Rules
3. Sum and Difference Rule.  For integrable 
functions f and g, 
4. Power Rule. For any real number n, 
where n ? -1, the indefinite integral x
n
of is,
1 2 1 2
[ ( ) ( )] ( ) ( ) f x f x dx f x dx f x dx
1
1
n
n
x
x dx C
n
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