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Unformatted text preview: ISE 4. Calculate
S OLUTION. Resist the urge to factor and instead complete the square:
Make the substitution so that , and so
Now, we are in a position to use the trigonometric substitution , so that , and
so
To return to our variable , we must construct the triangle generated by , yielding
. Then
where we absorb the constant into . Finally, converting back to our original variable gives
4 Improper Integrals (7.8) Previously, we dealt with integrals on a ﬁnite interval, with continuous integrands. An improper integral
is one that does not satisfy these conditions.
D EFINITION 4.1. Type I improper integrals.
(i) If exists for every number , then
provided this limit exists.
(ii) If exists for every number , then
provided this limit exists.
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WATERLOO SOS E XAM AID: MATH138 M IDTERM
The improper integrals and are called convergent if the corresponding limit exists
(ﬁnite is included in this deﬁnition) and divergent if the limit does not exist.
Finally, if both and are convergent, then we deﬁne
Here, any real number can be used (exercise 74 in section 7.8). E XAMPLE 8. Determine if the following integral converges, and if so, ﬁnd its value.
S OLUTION. By deﬁnition, if the integral converges, then
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This document was uploaded on 03/04/2014 for the course MATH 138 at Waterloo.
 Winter '07
 Anoymous
 Math, Calculus

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