Question

Which of the following is an improper integral?

Answer 1

Answer:

A)  $\displaystyle \int\limits^3_0 {\frac{x + 1}{3x - 2}} \, dx$

General Formulas and Concepts:

Calculus

Discontinuities

• Removable (Hole)
• Jump
• Infinite (Asymptote)

Integration

• Integrals
• Definite Integrals
• Integration Constant C
• Improper Integrals

Step-by-step explanation:

Let's define our answer choices:

A)  $\displaystyle \int\limits^3_0 {\frac{x + 1}{3x - 2}} \, dx$

B)  $\displaystyle \int\limits^3_1 {\frac{x + 1}{3x - 2}} \, dx$

C)  $\displaystyle \int\limits^0_{-1} {\frac{x + 1}{3x - 2}} \, dx$

D) None of these

We can see that we would have a infinite discontinuity if x = 2/3, as it would make the denominator 0 and we cannot divide by 0. Therefore, any interval that includes the value 2/3 would have to be rewritten and evaluated as an improper integral.

Of all the answer choices, we can see that A's bounds of integration (interval) includes x = 2/3.

∴ our answer is A.

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit:  Integration

Book: College Calculus 10e