Examples

Improper integral - infinite bound

Show that the improper integral converges. What is its value?

Solution

(1) Improper integral definition:

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(2) Replace infinity with a new symbol :

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(3) Take limit as :

Improper integral - infinite integrand

Show that the improper integral converges. What is its value?

Solution

(1) Improper integral definition:

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(2) Switch to and integrate:

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(3) Take limit as :

Example - Improper integral - infinity inside the interval

Does the integral converge or diverge?

Solution

(1) WRONG APPROACH:

It is tempting to compute the integral incorrectly, like this:

But this is wrong! There is an infinite integrand at . We must break it into parts!

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(2) Break apart at discontinuity:

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(3) Improper integral definition, both terms separately:

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(3) Integrate:

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(4) Limits:

Neither limit is finite. For to exist we’d need both of these limits to be finite. So the original integral diverges.

Comparison to p-integrals

Determine whether the integral converges:

(a)

(b)

Solution

(a) (1) Observe large tendency:

Consider large values. Notice the integrand tends toward for large .

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(2) Try comparison to :

Validate. Notice and when .

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(3) Apply comparison test:

We know:

We conclude:

(b) (1) Observe large tendency:

Consider large values. Notice the integrand tends toward for large .

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(2) Try comparison to :

Validate. Notice and when .

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(3) Apply comparison test:

We know:

We conclude: