# 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 ) = 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.