# Is this integral improper? If yes - why?

Is this integral improper? If yes - why? $$\int\limits^2_0 \,\frac{1}{x-1} dx$$

• at x=1 function $\dfrac{1}{x-1}\;$ is undefined Jun 10, 2013 at 13:48
• Hint: Check the definition of improper integral. Jun 10, 2013 at 14:21

Definition:

The integral $$\int_a^b f(x)dx$$ is called improper integral if:

• $$a=+\infty$$ or $$b=\infty$$ or both.

• $$f(x)$$ is unbounded at one or more points of $$a\le x\le b$$.

As @Git suggested verify which ones of above is satisfying the definition. You'll get the answer. ;-)

• Very nicely argued (just the right nudges needed!) +1 Jun 11, 2013 at 0:05
• @amWhy: Thanks my dear Amy. Yes indeed. :-) Jun 11, 2013 at 0:07

It is improper because the function "blows up" between the end points. That is, the function approaches $\pm \infty$ because the denominator is 0.

An improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.