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

Improper at both ends

Consider the improper integrals I=∫(π/2)πI = \int_(\pi/2)^\pi dx/sin⁡x/\sqrt{\sin x} and, for a≥0,Ia=∫a∞a \ge 0, I_{a} = \int_a^\infty dx/(x1+x2)/(x\sqrt{1 + x^{2}}). Which of the following statements are true?

  1. A.The integral I is convergent✓ Correct
  2. B.The integral I is not convergent
  3. C.The integral IaI_{a} converges for a = 1/2 but not for a = 0✓ Correct
  4. D.The integral IaI_{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, and why this one tests boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

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

More on this topic

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

From Integration › Improper integrals and convergence tests

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