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Techniques of Integration · Improper Integrals

Improper Integral: Type II (Discontinuity)

∫abf(x) dx=lim⁡t→a+∫tbf(x) dx (if f unbounded at a)\int_a^b f(x)\, dx = \lim_{t \to a^+} \int_t^b f(x)\, dx \text{ (if } f \text{ unbounded at } a\text{)}

When f has a discontinuity in [a,b], approach the discontinuity as a limit.

  • 2 worked examples
  • 16 in Techniques of Integration

Worked examples

Evaluate ∫₀¹ 1/√x dx.
  1. f is unbounded at x = 0. lim(t→0⁺) ∫ₜ¹ x^(-1/2) dx
  2. = lim(t→0⁺) [2√x]ₜ¹ = lim(t→0⁺) (2 - 2√t) = 2

Answer: 2 (converges)

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Evaluate ∫₀¹ 1/x² dx.

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Common questions

What is the Improper Integral: Type II (Discontinuity)?

When f has a discontinuity in [a,b], approach the discontinuity as a limit. It is one of the techniques of integration formulas in the CalcRef reference.

How do you use the Improper Integral: Type II (Discontinuity)?

Worked example. Evaluate ∫₀¹ 1/√x dx. f is unbounded at x = 0. lim(t→0⁺) ∫ₜ¹ x^(-1/2) dx. = lim(t→0⁺) [2√x]ₜ¹ = lim(t→0⁺) (2 - 2√t) = 2. Answer: 2 (converges).

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