# Thread: Converegence and divergence of integrals.

1. ## Converegence and divergence of integrals.

i have no idea to solve these problems. Can anyone give me the solutions? Thank you very much!!!

Explain why the following integrals are improper and determine whether they are
convergent or divergent. Evaluate those that are convergent.

a. {{{ int ( x/ root(x-1), dx, 1, 2 ) }}}
b. {{{ int ( 1/ (1+x^2), dx, -infinity, infinity ) }}}
c. {{{ int ( x(e^-3x), dx, 0, infinity ) }}}
d. {{{ int ( 1/ (x(ln^2)x), dx, pi, infinity ) }}}

2. For a) You can use the identity...

$\displaystyle \frac{x}{\sqrt{x-1}} = \sqrt{x-1} + \frac{1}{\sqrt{x-1}}$ (1)

... so that is...

$\displaystyle \int \frac{x}{\sqrt{x-1}}\ dx = \frac{2}{3}\ \sqrt{(x-1)^{3}} +2\ \sqrt{x-1} + c$ (2)

... and...

$\displaystyle \lim_{t \rightarrow 1} \int_{t}^{2} \frac{x}{\sqrt{x-1}}\ dx = \frac{8}{3}$ (3)

Kind regards

$\chi$ $\sigma$