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 right-hand 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 ) = 6x2 + 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.