Improper Integrals Assignment.sagews

Improper Integrals Assignment

Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

For each integral below:

• Graph the integrand (the function inside the integral).

• State why the integral is improper (there may be more than one reason).

• Set up an appropriate limit to compute the value of the integral, and calculate the limit.

• Evaluate the integral directly, and compare the answer with the limit.

Question 1

$\displaystyle\int_0^{\infty} \frac{1}{x^2+2x+4}\,dx \quad$ [Answer: $\frac{\pi\sqrt{3}}{9}$]

Question 2

$\displaystyle\int_0^1\frac{1}{\sqrt{x}}\,dx \quad$ [Answer: 2]

Question 3

$\displaystyle\int_0^{10}\frac{1}{ (x-5)^{2}}\,dx \quad$ [Answer: Divergent]

Question 4

$\displaystyle\int_0^{\infty}\frac{1}{e^x\sqrt{x}}$ [Answer: $\sqrt{\pi}$]

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
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