Let
𝑓
be a positive, continuous function with domain
𝑅
.

Assume
∫
1
∞
𝑓
(
𝑥
)
𝑑
𝑥
is convergent.

Which of the following improper integrals must also be convergent?

Select all the correct answers.

Question

Let 
𝑓
 be a positive, continuous function with domain 
𝑅
 .

Assume 
∫
1
∞
𝑓
(
𝑥
)
𝑑
𝑥
 is convergent.

Which of the following improper integrals must also be convergent?

Select all the correct answers.

BlackTom AI · Accepted Answer

Let me restate the problem and each option clearly to begin the analysis.
Question: Given a positive, continuous function f on R and that ∫ from 1 to ∞ of f(x) dx converges, which of the following improper integrals must also be convergent? Choose all correct answers.
Options:
1) ∫ from 1 to ∞ f(x) e^x dx
2) ∫ from 0 to ∞ f(x) dx
3) ∫ from 1 to ∞ f(x) dx
4) ∫ from 1 to ∞ f(x) sin^2 x dx
5) ∫ from 1 to ∞ -2 f(x) dx

Analysis of each option:
Option 1: ∫_1^∞ f(x) e^x dx
Think about how the factor e^x behaves. Since e^x grows exponentially and f(x) is only known to be positive and integrable on [1,∞) via ∫_1^∞ f(x) dx converging, there is no guarantee that the product f(x) e^x remains integrable. In fact, in general you can construct examples where f decays slowly enough that ∫_1^∞ f(x) dx converges but ∫_1^∞ f(x) e^x dx diverges. Therefore this integral is not guaranteed to converge and is not required to converge by the given information.

Option 2: ∫_0^∞ f(x) dx
Split this as ∫_0^1 f(x) dx + ∫_1^∞ f(x) dx. The first piece, ∫_0^1 f(x) dx, is over a finite interval of a continuous function, hence finite. The second piece, ∫_1^∞ f(x) dx, is given to converge. A sum of two finite numbers is finite, so ∫_0^∞ f(x) dx must converge. Therefore this integral is guaranteed to converge.

Option 3: ∫_1^∞ f(x) dx
This is exactly the given convergence condition. By hypothesis, this integral converges. So this must converge.

Option 4: ∫_1^∞ f(x) sin^2 x dx
Since 0 ≤ sin^2 x ≤ 1 for all x, we have f(x) sin^2 x ≤ f(x). Therefore, ∫_1^∞ f(x) sin^2 x dx is dominated by ∫_1^∞ f(x) dx, which is convergent. By comparison, ∫_1^∞ f(x) sin^2 x dx must converge. Hence this is guaranteed to converge.

Option 5: ∫_1^∞ -2 f(x) dx
This is just a constant multiple of ∫_1^∞ f(x) dx: ∫_1^∞ -2 f(x) dx = -2 ∫_1^∞ f(x) dx. Since the latter converges, multiplying by a finite constant -2 preserves convergence. Therefore this integral must converge.

Summary of conclusions (without stating a final single answer, per instruction):
- Option 1: Not guaranteed to converge; can fail.
- Option 2: Must converge.
- Option 3: Must converge (given).
- Option 4: Must converge (by comparison to a convergent integral).
- Option 5: Must converge (constant multiple of a convergent integral).

Let 𝑓 be a positive, continuous function with domain 𝑅 . Assume ∫ 1 ∞ 𝑓 ( 𝑥 ) 𝑑 𝑥 is convergent. Which of the following improper integrals must also be convergent? Select all the correct answers.Multiple choice

Similar Questions

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