Is the Integral of ln(x)/(x+2) from 0 to 5 an Improper Integral?

[ElectricMile]

the question states:

Is the integral, from 0 to 5, ln(x)/(x+2) dx, an improper integral? do not try to compute this integral.

the answer: Yes, ln(x) undifined when x = 0.

WHAT, why is this so, i thought an improper integral is when i take the upper/lower bound and replace it with infinity and take the limit? any help?

 

[benorin]

There are two catagories (three, if you count both as a separate category) of improper integrals: those that have infinite bounds, like

$$\int_{a}^{\infty} f(x)dx, \int_{-\infty}^{b} f(x)dx ,\mbox{ and } \int_{-\infty}^{\infty} f(x)dx,$$

and those which have integrands possessing infinite discountinuities somewhere in the domain of integration, and both. Yours is of the second variety.

 

[shmoe]

The label "improper integral" is attached to any definite integral where the function blows up or is otherwise undefined at the endpoints (points on the interval can be dealt with as well). In these cases the integral is defined as a limit, here:

$$\int_0^5 \frac{\log x}{x+2} dx=\lim_{\delta\rightarrow 0^+}\int_\delta^5 \frac{\log x}{x+2} dx$$