Originally Posted by

**^_^Engineer_Adam^_^** My hardest and last problem about I.I. :

0 has a discontinuity here so:

$\displaystyle \int_{0}^{1} \frac{dx}{x\sqrt{4-x^2}}$

$\displaystyle \lim_{t\to 0} \int_{t}^{1} \frac{dx}{x\sqrt{4-x^2}}$

to make it short, i used trigo substitution:

and im having trouble with this part

$\displaystyle \lim_{t \to 0} \frac{1}{2} \ln(\frac{2}{x} - \frac{\sqrt{4-x^2}}{x}) \big|_{t}^{1} $

$\displaystyle \lim_{t \to 0} \frac{1}{2} (\ln(2-\sqrt{4-x}) - \ln(x)) \big|_{t}^{1}$

$\displaystyle =\frac{1}{2} \ln(2-\sqrt{4-1}) - \frac{1}{2}\ln(1) - \frac{1}{2}\ln(2-2) + \frac{1}{2}\ln(0)$

which is

$\displaystyle \frac{1}{2} \ln(2-\sqrt{4-1}) - \frac{1}{2}\ln(1) - \infty + \infty$

i am having trouble because its indeterminate:

but i do not know how LHR works with ln

help

and thank you so much for the help on the other one