Part CCSIR NET June 2024improper-at-both-ends

Improper at both ends

Consider the improper integrals $I = \int_{(} π /2)^{π}$ dx $/ sin x$ and, for $a \geq 0, I_{a} = \int_{a}^{\infty}$ dx $/ (x 1 + x^{2})$ . Which of the following statements are true?

A.The integral I is convergent✓
B.The integral I is not convergent
C.The integral $I_{a}$ converges for a = 1/2 but not for a = 0✓
D.The integral $I_{a}$ converges for all a ≥ 0

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: If ∫₀^∞ f converges then f(x) → 0

The chapter behind this: Improper integrals — thresholds and tests — free to read

From Integration › Improper integrals and convergence tests

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