7.8 Integration Techniques

Improper Integrals

Question #4 (Medium): Convergent Improper Integral

Strategy

If both sides of the integral are infinity, then split the integral with respect to any number, but a nice

number would be beneficial in saving time for computation, like or . Then add limit with variable ,

since is already taken. Then solve the integral disregarding the limit, then afterward apply the value

that variable t approaches to determine convergence or divergence. If convergent, as limit value is

plugged in, it will figure itself out.

Sample Question

Determine if the integral is convergent or divergent. If convergent, evaluate the integral.

Solution

Since both sides approach infinity, split the integral. Since there is with no denomination, would be

good. Then:

. Add the limit:

. Then solve the function in the integral disregarding the limits for now.

The function follows after chain rule, so if comfortable solve as it is. If using substitution, let ,

then , then

. The interval has to align with the substitution. So when ,

, and when , . Because the integral is cut symmetrically, same info can be used

for both integrals.

Taking the first integral:

And the second integral:

Add the two with the limit:

They cancel out:

Therefore, the integral converges to

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###### Document Summary

If both sides of the integral are infinity, then split the integral with respect to any number, but a nice number would be beneficial in saving time for computation, like or . Then add limit with variable , since is already taken. Then solve the integral disregarding the limit, then afterward apply the value that variable t approaches to determine convergence or divergence. If convergent, as limit value is plugged in, it will figure itself out. Determine if the integral is convergent or divergent. Since both sides approach infinity, split the integral. Since there is with no denomination, would be good. The function follows after chain rule, so if comfortable solve as it is. Then solve the function in the integral disregarding the limits for now. The interval has to align with the substitution. Because the integral is cut symmetrically, same info can be used for both integrals.

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