Question

Explain why each of the following integrals is improper. (a) \int_6^7 \frac{x}{x-6}dx -Since the integral has an infinite interval of integration, it

Applications of integrals
ANSWERED
asked 2021-06-12
Explain why each of the following integrals is improper.
(a) \(\int_6^7 \frac{x}{x-6}dx\)
-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.
-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.
-The integral is a proper integral.
(b)\(\int_0^{\infty} \frac{1}{1+x^3}dx\)
Since the integral has an infinite interval of integration, it is a Type 1 improper integral.
Since the integral has an infinite discontinuity, it is a Type 2 improper integral.
The integral is a proper integral.
(c) \(\int_{-\infty}^{\infty}x^2 e^{-x^2}dx\)
-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.
-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.
-The integral is a proper integral.
d)\(\int_0^{\frac{\pi}{4}} \cot x dx\)
-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.
-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.
-The integral is a proper integral.

Expert Answers (1)

2021-06-13
a)
Consider the integral \(\int_6^7 \frac{x}{x-6}dx\)
It is improper because it is discontinuous at x=6
Since the integral has an infinite discontinuity , it is a type 2 improper integral.
b) Let us consider the integral \(\int_0^{\infty} \frac{dx}{1+x^3}\)
Since the integral has an infinite interval of integration. That is \((0,\infty)\)
It is Type 1 improper integral.
c)
Consider the integral \(\int_{-\infty}^{\infty} x^2 e^{-x^2}dx\)
Since the integral has an infinite interval of integration. That is \((-\infty,\infty)\)
It is Type 1 improper integral.
d)
Consider the interval \(\int_0^{\frac{\pi}{4}} \cot x dx\)
It is improper because it is discontinuous at x=0
Since the integral has an infinite discontinuity , it is a type 2 improper integral.
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