Which of the following integrals are IMPROPER?

Select all the correct answers.

Question

Which of the following integrals are IMPROPER?

Select all the correct answers.

BlackTom AI · Accepted Answer

Let’s parse the question carefully: we are asked which of the listed integrals are improper, and to select all that apply. An improper integral is one where either the interval of integration is infinite, or the integrand has a singularity (becomes unbounded) at some point in the interval.

Option 1: ∫ from 1 to ∞ of x dx. Here the interval extends to infinity, so the integral is considered improper due to the infinite limit. The integrand x is continuous on (1, ∞), but the improper nature arises from the unbounded interval, requiring a limit process to define convergence.

Option 2: ∫ from −∞ to ∞ of x dx. This one has both ends extending to infinity. Any integral with infinite limits can be improper, and in this case, the integral does not converge in the usual sense because the area grows without bound in both directions. Therefore, this is improper.

Option 3: ∫ from 1 to 5 of (1/x) dx. The interval is finite and the integrand 1/x is continuous on [1,5], so there is no singularity and no infinite limit. This is a proper (finite) definite integral.

Option 4: ∫ from 0 to 1 of x dx. The interval is finite and the integrand x is continuous on [0,1], so this is also a proper integral. There is no singularity at 0 since x is well-behaved there.

Option 5: ∫ from 0 to 1 of (1/x) dx. The integrand 1/x has a singularity at x = 0, which lies inside the interval [0,1]. This makes the integral improper due to the unbounded integrand near 0, and one would evaluate it via a limit as x approaches 0+.

To summarize the reasoning for each option, we see that options 1, 2, and 5 involve either infinite limits or a singular integrand within the interval, which makes them improper. Options 3 and 4 are proper integrals with finite limits and well-behaved integrands.

In the provided answer set, the improper integrals are those corresponding to options 1, 2, and 5. The finite-interval, continuous cases correspond to options 3 and 4. This matches the notion that improper integrals arise from unbounded domains or integrands with singularities inside the interval.

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 54 (12.1-12.6)

Which of the following integrals are IMPROPER? Select all the correct answers.

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