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).

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 55(12.7,12.8)

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.

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类似问题

Let 𝑓 , 𝑔 be positive, continuous functions with domain 𝑅 . Let 𝐿 = lim 𝑥 → ∞ 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) . Assume ∫ 1 ∞ 𝑓 ( 𝑥 ) 𝑑 𝑥 is convergent. Which of the following statements must be true? Select all the correct answers.

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

Consider an improper integral ∫ 1 ∞ 1 𝑥 2 𝑎 − 1 𝑑 𝑥 . Which condition should 𝑎 satisfy to make this improper integral convergent? Select the best answer (which includes as many values as possible).

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

Which of the following statements is correct? Note: only one of them is correct.

In Question 0, select all of the integrals that DIVERGE.

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