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Improper integrals of the first kind



 

Def. 1.Let a function is continuous on an interval . If there exists a finite limit

( 6 )

then we say that the next integral (an improper integral of the first kind, an integral with infinite upper limit)

( 7 )

converges(exists, is convergent).

Thus by the definition 1

. ( 8 )

Def. 2.If the limit (6) is infinite one or doesn’t exist we say that the improper integral (7) diverges (doesn’t exist, is divergent).

By the same way we can define the next two improper integrals of the first kind

Def. 3. ( 9 )

if a function is continuous on an interval .

Def. 4. . ( 10 ) if a function is continuous on the set of all reals.

Integral (9) is called convergent if the limit in (9) exists and divergent otherwise. The same is concerned to the integral (10).

Def. 5.Principle value of the improper integral (10) is called the next limit

. ( 11 )

If an integral (10) converges then its principal value also converges. But there are cases when the integral (10) diverges but its principal value converges.

Ex. 2. Improper integrals

( 12 )

are convergent for and divergent for .

■Let’s consider the first integral.

a) If we can suppose where , and so

,

that is the integral converges for .

b) Let In this case

.

The integral diverges.

c) If we put where , and

.

The integral diverges.■

Ex. 3. Prove that the integral diverges but its principal value converges.

■On the base of the formula (10)

Both limits doesn’t exist and therefore the integral diverges. On the other hand the principal value of the integral converges to zero (or equals zero), because of by the formula (11)

.■

Ex. 4. Find the area of an infinite figure bounded by Agnesi witch and its asymptote (fig. 6).

The straight line (the Ox-axis) is a horizontal
Fig. 6 asymptote of Agnesi witch because of

.

The figure is symmetric with respect to the -axis, and therefore its area equals

.





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