Integral is improper if we have to take limit in order to calculate it. This can happen if we have infinite values of integration or if the interval if integration contains point(s) where the function is not defined. The latter is the case here because the function we are integrating is not defined for x=1.
Because this point is within the interior of the interval of integration (not at the endpoint) we first must write this integral as a sum of two integrals.
int_0^2 1/(x-1)^2 dx=int_0^1 1/(x-1)^2dx+int_1^2 1/(x-1)^2dx=
Substitute u=x-1 => du=dx. Since u=x-1 all bounds of integration become lower by 1.
int_-1^0 1/u^2 du+int_0^1 1/u^2 du=-1/u|_-1^0-1/u|_0^1=
lim_(u to 0^-) -1/u+1/-1-1/1+lim_(u to 0^+) 1/u=
Notice the use of directional limits (from the left for the first and from the right for the second integral).
-(-infty)-2+infty=infty
As we can see the integral diverges.
The image below shows the graph of the function and area under it corresponding to the value of the integral. Asymptotes of the graph are x-axis and line x=1. The other image (the red one) shows the graph of the function f(u)=1/u. There we see why the two directional limits have different values.
https://en.wikipedia.org/wiki/Improper_integral
Thursday, August 22, 2019
int_0^2 1/(x-1)^2 dx Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges
Subscribe to:
Post Comments (Atom)
Why is the fact that the Americans are helping the Russians important?
In the late author Tom Clancy’s first novel, The Hunt for Red October, the assistance rendered to the Russians by the United States is impor...
-
There are a plethora of rules that Jonas and the other citizens must follow. Again, page numbers will vary given the edition of the book tha...
-
The only example of simile in "The Lottery"—and a particularly weak one at that—is when Mrs. Hutchinson taps Mrs. Delacroix on the...
-
A good thesis statement presents a claim (an interpretive stance on a story that can be defended using textual evidence) and is a position w...
-
The given two points of the exponential function are (2,24) and (3,144). To determine the exponential function y=ab^x plug-in the given x an...
-
What does the hot air balloon symbolize? To the Assad son who buys the hot air balloon, it symbolizes a kind of whimsy that he can afford. B...
-
The play Duchess of Malfi is named after the character and real life historical tragic figure of Duchess of Malfi who was the regent of the ...
-
Allie’s baseball mitt is extremely important to Holden in The Catcher in the Rye. It is a symbol of Allie since it was important to his brot...
No comments:
Post a Comment