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Explain why each of the following integrals is improper.

  • (a) 23xx2dx\int_{2}^{3} \frac{x}{x - 2} \, dx
  • (b) 011+x3dx\int_{0}^{\infty} \frac{1}{1 + x^3} \, dx
  • (c) x2ex2dx\int_{-\infty}^{\infty} x^2 e^{-x^2} \, dx
  • (d) 0π/4cot(x)dx\int_{0}^{\pi/4} \cot(x) \, dx

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Question :

Explain why each of the following integrals is improper.

  • (a) 23xx2dx\int_{2}^{3} \frac{x}{x - 2} \, dx
  • (b) 011+x3dx\int_{0}^{\infty} \frac{1}{1 + x^3} \, dx
  • (c) x2ex2dx\int_{-\infty}^{\infty} x^2 e^{-x^2} \, dx
  • (d) 0π/4cot(x)dx\int_{0}^{\pi/4} \cot(x) \, dx

![Explain why each of the following integrals is improper.

Solution:

Neetesh Kumar

Neetesh Kumar | December 10, 2024

Calculus Homework Help

This is the solution to Math 132
Assignment: 7.8 Question Number 1
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Step-by-step solution:

(a) Improper Integral Explanation

The given integral is: 23xx2dx\int_{2}^{3} \frac{x}{x - 2} \, dx

  • The denominator x2x - 2 becomes 00 at x=2x = 2, which is the lower limit of integration.
  • This creates an infinite discontinuity within the range of integration.

Reason: Since the integral has an infinite discontinuity at the lower limit x=2x = 2, it is classified as a Type 2 improper integral.

(b) Improper Integral Explanation

The given integral is: 011+x3dx\int_{0}^{\infty} \frac{1}{1 + x^3} \, dx

  • The upper limit of integration is x=x = \infty, which makes the interval of integration infinite.

Reason: Since the integral has an infinite interval of integration, it is classified as a Type 1 improper integral.

(c) Improper Integral Explanation

The given integral is: x2ex2dx\int_{-\infty}^{\infty} x^2 e^{-x^2} \, dx

  • The limits of integration are -\infty and \infty, which means the interval of integration is infinite.

Reason: Since the integral has an infinite interval of integration, it is classified as a Type 1 improper integral.

(d) Improper Integral Explanation

The given integral is: 0π/4cot(x)dx\int_{0}^{\pi/4} \cot(x) \, dx

  • The integrand cot(x)\cot(x) has a discontinuity at x=0x = 0, which is the lower limit of integration.

Reason: Since the integral has an infinite discontinuity at the lower limit x=0x = 0, it is classified as a Type 2 improper integral.

Final Classification Summary:

(a): Type 2 improper integral (infinite discontinuity at x=2){\boxed{\text{Type 2 improper integral (infinite discontinuity at} \space x = 2)}}

(b): Type 1 improper integral (infinite interval of integration)\boxed{\text{Type 1 improper integral (infinite interval of integration)}}

(c): Type 1 improper integral (infinite interval of integration)\boxed{\text{Type 1 improper integral (infinite interval of integration)}}

(d): Type 2 improper integral (infinite discontinuity at x=0){\boxed{\text{Type 2 improper integral (infinite discontinuity at} \space x = 0)}}


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