Integral with variable upper limit of integration
Let f ( x) be a continuous function, given in a
segment [ a , b ], then for any x
[ a , b ] the function
exists. This function is given as an integral with variable upper limit of integration
in the righthand part of the equality.
All rules and properties of a definite integral apply to an integral with variable upper limit of integration.
E x a m p l e . 
A variable force acting on a linear way changes in the law: f (
x ) = 6x^{2} + 5 at x
0 . What law does a work of this force change in ? 
S o l u t i o n. 
A work of the force f ( x ) on a segment [ 0 , x ] of linear way is equal to:
Thus, the work changes in
the law: F ( x) = 2x^{ 3} + 5x. 
According to the definition of an integral with variable upper limit of integration or the
function F ( x ) and known properties of an integral it follows that at x
[ a , b]
F' ( x ) = f ( x ) .
Check this property using the above mentioned example.
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