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# PPT: Integral Calculus GATE Notes | EduRev

## GATE : PPT: Integral Calculus GATE Notes | EduRev

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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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