Sum rule in integration
Encyclopedia
In calculus
, the sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals. It is of particular use for the integration
of sum
s, and is one part of the linearity of integration
.
As with many properties of integrals in calculus, the sum rule applies both to definite integrals and indefinite integrals. For indefinite integrals, the sum rule states
of exp(x) is exp(x) from calculus with exponentials and that the integral
of cos(x) is sin(x) from calculus with trigonometry then:
Some other general results come from this rule. For example:
The proof above relied on the special case of the constant factor rule in integration with k=-1.
Thus, the sum rule might be written as:
Another basic application is that sigma and integral signs can be changed around. That is:
This is simply because:
Since the integral is similar to a sum anyway, this is hardly surprising.
disappears).
as the antiderivative
, the reverse process of differentiation
:
Adding
these,
Now take the sum rule in differentiation
:
Integrate both sides with respect to x:
So we have, looking at (1) and (2):
Therefore:
Now substitute:
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, the sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals. It is of particular use for the integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of sum
SUM
SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
s, and is one part of the linearity of integration
Linearity of integration
In calculus, linearity is a fundamental property of the integral that follows from the sum rule in integration and the constant factor rule in integration. Linearity of integration is related to the linearity of summation, since integrals are thought of as infinite sums.Let ƒ and g be functions...
.
As with many properties of integrals in calculus, the sum rule applies both to definite integrals and indefinite integrals. For indefinite integrals, the sum rule states
Application to indefinite integrals
For example, if you know that the integralIntegral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of exp(x) is exp(x) from calculus with exponentials and that the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of cos(x) is sin(x) from calculus with trigonometry then:
Some other general results come from this rule. For example:
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The proof above relied on the special case of the constant factor rule in integration with k=-1.
Thus, the sum rule might be written as:
Another basic application is that sigma and integral signs can be changed around. That is:
This is simply because:
Since the integral is similar to a sum anyway, this is hardly surprising.
Application to definite integrals
Passing from the case of indefinite integrals to the case of integrals over an interval [a,b], we get exactly the same form of rule (the arbitrary constant of integrationArbitrary constant of integration
In calculus, the indefinite integral of a given function is only defined up to an additive constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives...
disappears).
The proof of the rule
First note that from the definition of integrationIntegral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
as the antiderivative
Antiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
, the reverse process of differentiation
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
:
Adding
SUM
SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
these,
Now take the sum rule in differentiation
Sum rule in differentiation
In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. This is a part of the linearity of differentiation. The sum rule in integration follows from it...
:
Integrate both sides with respect to x:
So we have, looking at (1) and (2):
Therefore:
Now substitute: