Question

Which of the following is an improper integral?

A) the integral from 0 to 3 of the quotient of the quantity x plus 1 and the quantity 3 times x minus 2 dx
B) the integral from 1 to 3 of the quotient of the quantity x plus 1 and the quantity 3 times x minus 2 dx
C) the integral from negative 1 to 0 of the quotient of the quantity x plus 1 and the quantity 3 times x minus 2 dx
D) None of these

Answer 1

A)
`\displaystyle \int\limits^3_0 {(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

Explanation:

Let's define our answer choices:

A)
`\displaystyle \int\limits^3_0 {(x + 1)/(3x - 2)} \, dx`

B)
`\displaystyle \int\limits^3_1 {(x + 1)/(3x - 2)} \, dx`

C)
`\displaystyle \int\limits^0_(-1) {(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