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.

Question

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.

BlackTom AI · Accepted Answer

Question restatement and options:
- Given a positive, continuous function f with domain R, and knowing that ∫ from 1 to ∞ of f(x) dx diverges, determine which of the following improper integrals must also be divergent. Select all that apply.
Options:
1) ∫_0^∞ f(x) dx
2) ∫_1^∞ f(x) /(1+x) dx
3) ∫_1^∞ f(x)/(1+sin^2 x) dx
4) ∫_1^∞ f(x)^(1+sin^2 x) dx
5) ∫_1^∞ f(x)^2 dx

Option-by-option analysis:
Option 1: ∫_0^∞ f(x) dx
Since f is positive and continuous on R, the integral over [0,1] is finite (a continuous function on a compact interval has a finite integral). The divergence of ∫_1^∞ f(x) dx implies that the tail integral from 1 to ∞ is infinite. Adding the finite part over [0,1] to an infinite tail cannot cancel the divergence. Therefore, ∫_0^∞ f(x) dx must diverge. This reasoning relies on the fact that the sum of a finite number and an infinite quantity is infinite.

Option 2: ∫_1^∞ f(x)/(1+x) dx
Here, for x ≥ 1, we have 1+x ≥ x, so 0 < 1/(1+x) ≤ 1/x, and in particular 1/(1+x) ≤ 1. The integrand is f(x) times a bounded factor 1/(1+x) that decays like 1/x. However, divergence of ∫_1^∞ f(x) dx does not automatically imply divergence of ∫_1^∞ f(x)/(1+x) dx, because multiplying by a decaying factor can sometimes produce a convergent integral even if the original diverges. Hence, this option need not diverge in all cases, and cannot be declared must-diverge.

Option 3: ∫_1^∞ f(x)/(1+sin^2 x) dx
Note that 1 ≤ 1+sin^2 x ≤ 2 for all x, so 1/(1+sin^2 x) satisfies 1/2 ≤ 1/(1+sin^2 x) ≤ 1. Consequently, f(x)/(1+sin^2 x) ≥ (1/2) f(x) for all x. Since ∫_1^∞ f(x) dx diverges and (1/2) f(x) is just a constant multiple of f(x), the comparison yields ∫_1^∞ f(x)/(1+sin^2 x) dx ≥ (1/2) ∫_1^∞ f(x) dx = ∞. Therefore, this integral must diverge.

Option 4: ∫_1^∞ f(x)^(1+sin^2 x) dx
The exponent 1+sin^2 x lies between 1 and 2. Depending on the size of f(x) relative to 1, raising to an exponent between 1 and 2 can either increase or decrease the value of f(x). Without additional information about the magnitude of f(x) (whether it’s consistently ≥1 or ≤1 for large x), we cannot guarantee divergence here. A counterexample can be constructed where ∫_1^∞ f(x) dx diverges but ∫_1^∞ f(x)^(1+sin^2 x) dx converges, so this option is not necessarily must-diverge.

Option 5: ∫_1^∞ f(x)^2 dx
Similar to option 4, squaring f can either amplify or dampen the integrand depending on the size of f(x). It is possible to have ∫_1^∞ f(x) dx diverge while ∫_1^∞ f(x)^2 dx converges (for example, a tail like f(x) ~ 1/x yields divergence for ∫ f but convergence for ∫ f^2). Therefore, this integral is not guaranteed to diverge.

Summary of findings:
- Must-diverge: Option 1 (∫_0^∞ f(x) dx) and Option 3 (∫_1^∞ f(x)/(1+sin^2 x) dx).
- Not necessarily must-diverge: Options 2, 4, and 5.

Therefore, the correct selections are the first and the third options.

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 divergent. Which of the following improper integrals must also be divergent? 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 convergent. Which of the following improper integrals must also be convergent? 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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