- Sum rule in integration
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 ofsum s, and is one part of thelinearity of integration .As with many properties of integrals in calculus, the sum rule applies both to
definite integral s andindefinite integral s. For indefinite integrals, the sum rule states:
Application to indefinite integrals
For example, if you know that the
integral of exp(x) is exp(x) fromcalculus with exponentials and that theintegral of cos(x) is sin(x) fromcalculus 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.
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 integration disappears).The proof of the rule
First note that from the definition of integration 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:
::
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