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8.8 Improper Integrals. Until now we have been finding integrals of continuous functions over closed intervals. Sometimes we can find integrals for functions where the function is discontinuous or the limits are infinite. These are called improper integrals .

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8.8

Improper Integrals


Until now we have been finding integrals of continuous functions over closed intervals.

Sometimes we can find integrals for functions where the function is discontinuous or the limits are infinite. These are called improper integrals.


I when the limit of integration is infinite
I. When the limit of integration is infinite

  • Consider

  • We calculate

  • Now we take the limit as b∞

  • So we say convergesto 1


Ii when the integrand becomes infinite
II. When the integrand becomes infinite

  • Consider

  • In this case we may have a finite interval, but the function may be unbounded somewhere on the interval since it has a vertical asymptote at x = 0

  • We compute

    Now we take the limit

  • So converges to 2


Example

(right hand limit)

We approach the limit from inside the interval.

This integral diverges.


Example

The function is undefined at x = 1 .

Vertical asymptote at x= 1

(left hand limit)

We must approach the limit from inside the interval.


This integral converges.


If then gets bigger and bigger as , therefore the integral diverges.

If then b has a negative exponent and , therefore the integral converges.

Example

(P is a constant.)

What happens here?


Recall
Recall , therefore the integral

If either of the integrals diverges, the whole thing diverges


Example , therefore the integral


Examples , therefore the integral


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