Q: Which condition must be satisfied for the Limit Comparison Test to be applied to $$\int_a^\infty f(x) \, dx$$ and $$\int_a^\infty g(x) \, dx$$?

$$\lim_{x \to \infty} \frac{f(x)}{g(x)} = L$$, $$0 \u003c L \u003c \infty$$

Q: Which of the following is the correct way to express $$\int_a^\infty f(x) \, dx as a $$limit?

$$\int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx$$

Q: Which of the following is a necessary step when evaluating $$\$$int_0$$^1 \frac{dx}{(x-1$$)^{2/3}$$}$$?

Express the integral as a limit as $$b \to 1$$^{-}$$.

Question 1

Which of the following is an example of an improper integral of Type I?

Accepted Answer

$$\$$int_1$$^\infty e$$^{-x}$$ \, dx$$

Question 2

Evaluate $$\int_{-\infty}^{\infty} \frac{dx}{1 + x^2$}.$$

Accepted Answer

$$\pi$$

Question 3

Which condition must be satisfied for the Limit Comparison Test to be applied to $$\int_a^\infty f(x) \, dx$$ and $$\int_a^\infty g(x) \, dx$$?

Accepted Answer

$$\lim_{x 	o \infty} \frac{f(x)}{g(x)} = L$$, $$0 \u003c L \u003c \infty$$

Question 4

Suppose $$\$$int_1$$^\infty \frac{dx}{x^p}$$ converges. What must be true about $$p$$?

Accepted Answer

$$p \u003e 1$$

Question 5

Which of the following integrals is improper due to a discontinuity at an endpoint?

Accepted Answer

$$\$$int_0$$^1 \frac{1}{\sqrt{x}} \, dx$$

Question 6

Evaluate $$\$$int_0$$^3 \frac{dx}{(x-1$$)^{2/3}$$}.$$

Accepted Answer

$$3 + 3 \sqrt[3]{2}$$

Question 7

If $$\$$int_1$$^\infty g(x) \, dx$$ diverges and $$0 \leq f(x) \leq g(x)$$ for all $$x \geq 1$$, what can you conclude about $$\$$int_1$$^\infty f(x) \, dx$$?

Accepted Answer

It diverges.

Question 8

Which of the following is the correct way to express $$\int_a^\infty f(x) \, dx as a $$limit?

Accepted Answer

$$\int_a^\infty f(x) \, dx = \lim_{b 	o \infty} \int_a^b f(x) \, dx$$

Question 9

Given $$\$$int_0$$^\infty e$$^{-x^2}$$ \, dx$$ and $$\$$int_0$$^\infty e$$^{-x}$$ \, dx$$, which comparison test can be used to determine convergence?

Accepted Answer

Direct Comparison Test

Question 10

Suppose $$f(x) is $$continuous on $$(-\infty, \infty). $$Which is a valid way to write $$\int_{-\infty}^{\infty} f(x) \, dx$$?

Accepted Answer

$$\int_{-\infty}^{\infty} f(x) \, dx = \int_{-\infty}^c f(x) \, dx + \int_c$$^{\infty}$$ f(x) \, dx$$

Question 11

Evaluate $$\$$int_1$$^\infty \frac{\ln x}{x^2$} \, dx.$$

Accepted Answer

$$1$$

Question 12

Which of the following statements about improper integrals is FALSE?

Accepted Answer

Improper integrals always diverge.

Question 13

If $$\$$int_1$$^\infty \frac{dx}{x^p}$$ diverges, what is the range of $$p$$?

Accepted Answer

$$p \u003c 1$$

Question 14

Which of the following is a necessary step when evaluating $$\$$int_0$$^1 \frac{dx}{(x-1$$)^{2/3}$$}$$?

Accepted Answer

Express the integral as a limit as $$b 	o 1$$^{-}$$.

Question 15

Which theorem allows you to conclude that if $$\int_a^\infty g(x) \, dx$$ converges and $$0 \leq f(x) \leq g(x)$$ for all $$x \geq a$$, then $$\int_a^\infty f(x) \, dx$$ also converges?

Accepted Answer

Direct Comparison Test

Question 16

Suppose $$f(x) is $$discontinuous at $$c in$$ $$[a, b]$$ but continuous elsewhere. How can $$\int_a^b f(x) \, dx be $$evaluated?

Accepted Answer

$$\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$$

Improper Integrals (Section 8.7): Definitions, Types, and Convergence Tests

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