Unizor - Definite Integrals - Improper Integrals Examples 1

Notes to a video lecture on http://www.unizor.com

Improper Definite Integrals

Example 1.1

∫₀¹1/√x dx =
= lim_a→0 ∫_a¹1/√x dx
Indefinite integral of f(x)=1/√x is F(x)=2√x.
Indeed, let's take a derivative of F(x)=2√x:
D_x F(x) = 2·(1/2)·x^(1/2)−1 =
= x^−1/2 = 1/√x

Using Newton-Leibniz formula,
∫_a¹1/√x dx = F(1) − F(a) =
= 2√1 − 2√a
As a→0, this expression converges to 2

Answer: 2

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Example 1.2

∫₀^∞e^−x dx =
= lim_b→∞ ∫₀^be^−x dx
Indefinite integral of f(x)=e^−x is F(x)=−e^−x.
Evaluating integral:
∫₀^be^−x dx = F(b) − F(0) =
= (−e^−b) − (−e⁻⁰) = 1 −e^−b
As b→∞, this expression converges to 1

Answer: 1

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Example 1.3

∫₀^∞1/(1+x²) dx =
= lim_b→∞ ∫₀^b1/(1+x²) dx
Indefinite integral of f(x)=1/(1+x²) is F(x)=arctan(x).
Evaluating integral:
∫₀^b1/(1+x²) dx = F(b) − F(0) =
= arctan(b) − arctan(0) =
= arctan(b)
As b→∞, this expression converges to π/2

Answer: π/2

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Example 1.4

∫₁^∞(1/x²)·e^1/x dx =
= lim_b→∞ ∫₁^b(1/x²)·e^1/x dx
Indefinite integral of
f(x)=(1/x²)·e^1/x can be found by noticing that −1/x² is a derivative of 1/x and, therefore, a derivative of e^1/x is e^1/x·(−1/x²).
Therefore, indefinite integral of f(x)=(1/x²)·e^1/x is F(x)=−e^1/x.
Evaluating integral:
∫₁^b(1/x²)·e^1/x dx = F(b)−F(1) =
= (−e^1/b) − (−e^1/1) = e − e^1/b
As b→∞, this expression converges to e −1

Answer: e −1