We study the concept of improper integrals.
- OpenStax II 3.7: Improper Integrals
- Ximera OSU: Improper Integrals
- Community Calculus: Improper Integrals
- First off, we note that the integral is in fact improper because has a vertical asymptote at . We can find the antiderivative of by integration by parts: (your answer should be the usual thing given by integration by parts. In particular, it equals at .)
- The correct interpretation of the improper integral is
- Using the antiderivative you found above (with ), use the Fundamental Theorem of Calculus to conclude
- As , and . Using knowledge of orders of growth, we know that the product tends to because grows faster than any negative power of as . tends to some finite constant as because has the same growth rate as as . tends to zero because grows slower than any negative power of as .
- We take the limit as to conclude (write N/A if the limit diverges).