Improper Integrals

Math ⇒ Calculus

Improper Integrals starts at 12 and continues till grade 12. QuestionsToday has an evolving set of questions to continuously challenge students so that their knowledge grows in Improper Integrals. How you perform is determined by your score and the time you take. When you play a quiz, your answers are evaluated in concept instead of actual words and definitions used.

See sample questions for grade 12

Describe the difference between an improper integral of the first kind and the second kind.

Describe the process for evaluating an improper integral with an infinite upper limit.

Does the integral ∫₁^∞ 1/x dx converge or diverge?

Explain why the integral ∫₀¹ 1/√x dx is improper.

Explain why the integral ∫₀¹ 1/x^p dx diverges for p ≥ 1.

Explain why the integral ∫₁^∞ 1/x² dx converges but ∫₁^∞ 1/x dx diverges.

State the comparison test for improper integrals.

State the definition of an improper integral with an infinite limit of integration.

Which of the following integrals converges? (1) ∫₁^∞ 1/x dx (2) ∫₁^∞ 1/x² dx (3) ∫₀¹ 1/x dx (4) ∫_-1¹ 1/x dx

Which of the following integrals diverges? (1) ∫₁^∞ 1/x² dx (2) ∫₀¹ 1/x dx (3) ∫₀¹ x dx (4) ∫₀^∞ e^-x dx

Which of the following integrals is improper due to an infinite interval? (1) ∫₀¹ 1/x dx (2) ∫₁^∞ 1/x dx (3) ∫_-1¹ 1/x dx (4) ∫₀¹ x dx

Which of the following integrals is improper? (1) ∫₀² x dx (2) ∫₁^∞ e^-x dx (3) ∫_-1¹ 1/x dx (4) Both (2) and (3)

Fill in the blank: The integral ∫₀¹ 1/x^p dx converges for p < _______.

Fill in the blank: The integral ∫₀¹ 1/x^p dx converges if and only if p < _______.

Fill in the blank: The integral ∫₁^∞ 1/x^p dx converges for p _______ 1.

Fill in the blank: The integral ∫₁^∞ 1/x^p dx converges if and only if p > _______.

If ∫₁^∞ f(x) dx converges, must f(x) approach zero as x approaches infinity?

True or False: The integral ∫₀^∞ 1/(x²+1) dx converges.

True or False: The integral ∫₀^∞ e^-x dx converges.

True or False: The integral ∫₀^∞ x e^-x dx converges.