Table of contents |
2 Application to definite integrals 3 The proof of the rule |
For example, if you know that the integral 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:
Application to indefinite integrals
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:
Passing rom 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).
First note that from the definition of integration as the antiderivative, the reverse process of differentiation:
Application to definite integrals
The proof of the rule
Adding these,
Now take the sum rule in differentiation: