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.

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