\input amstex
\input cyracc.def
\font\tencyr=wncyr10
\def\cyr#1{{\tencyr\cyracc#1}}
\font\tencyri=wncyi10
\def\cyri#1{{\tencyri\cyracc#1}}
\font\tencyb=wncyb10
\def\cyrb#1{{\tencyb\cyracc#1}}

{\bf 6.19} \cyri{Dokazati jednakost}
$$\int_0^{\frac{\pi}{2}}\frac{\text{d}x}{\sqrt{\tan x}}=%
\int_0^{\frac{\pi}{2}}\sqrt{\tan x}\,\text{d}x=\frac{\pi}{\sqrt{2}}.$$

\cyrb{Reshenje.} \cyr{Oba integrala su nesvojstvena. Prvi ima tachku}
$x=0$ \cyr{za singularitet, a drugi tachku} $x=\frac{\pi}{2}$.
\cyr{S obzirom da je}
$$\lim_{x\rightarrow\frac{\pi}{2}^-}\frac{1}{\sqrt{\tan x}}=0,$$
\cyr{tachka} $\frac{\pi}{2}$ \cyr{je prividni singularitet prvog
integrala. Smenom} $\tan x=y^2$, \cyr{ili shto je ekvivalentno
sa} $x=\arctan y^2$, \cyr{interval} $[0,+\infty)$ \cyr{preslikava
se na poluzatvoreni interval} $[0,\frac{\pi}{2})$. \cyr{Dakle,
is\-pu\-nje\-ni su svi uslovi o smeni pro\-men\-lji\-ve u nesvojstvenom
integralu.} $\ldots$
\bye