This theorem is divided in two parts:
1. First fundamental theorem of Calculus
The derivative of the integral of a function f with respect to the variable t over the interval [a, x] is
equal to the function f with respect to x. This is expressed by:
This theorem shows the relationship between the integral and the derivative. It tells us that the integral of a function f can be calculated by taking its integral over a variable bound of integration.meaning that the upper limit of integration is variable. In fact the above relationship shows that when we take the derivative of the integral we find the initial function f.
2. Second fundamental theorem of Calculus
The definite integral of a function f over an interval [a,b] is equal to the antiderivative at b minus
the antiderivative at a. If a function f is continuous over an interval [a,b] and F is an antiderivative
of f over this interval, then:
This theorem shows the relationship between the definite integral and the indefinite integral.
Practice
1. Evaluate:
Solution
2. Use the fundamental theorem of Calculus to find the derivative of
N.B. In this exercise it's dt instead of dx.
Solution
Exercise
Find the derivative of:
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