Question

integrate the function 1/(x - √x).dx

Answer 1

given, $\int{\frac{1}{x-\sqrt{x}}}\,dx$

=$\int{\frac{1}{\sqrt{x}(\sqrt{x}-1)}}\,dx$
(√x - 1) = t
differentiate both sides,
1/2√x dx = dt
dx/√x = 2dt , put it in above integration

so, $\int{\frac{1}{\sqrt{x}(\sqrt{x}-1)}}\,dx$

= $\int{\frac{1}{t}}\,2dt$

= $2\int{\frac{dt}{t}}$
= 2log|t| + C
put t = (√x - 1)
so, = 2log|(√x - 1)| + C

Answer 2

HELLO DEAR,

given, $\bold{\int{\frac{1}{x-\sqrt{x}}}\,dx}$

$\bold{\int{\frac{1}{\sqrt{x}(\sqrt{x}-1)}}\,dx}$ 
let (√x - 1) = t 
differentiate both sides, 
$\Rightarrow$ 1/2√x dx = dt 
dx/√x = 2dt ,

put it in integration

so, $\bold{\int{\frac{1}{\sqrt{x}(\sqrt{x}-1)}}\,dx}$

 $\bold{\int{\frac{1}{t}}\,2dt}$

$\bold{2\int{\frac{dt}{t}}}$

= 2log|t| + C 

put t = (√x - 1) 

so, = 2log|(√x - 1)| + C

whew, c is the arbitrary constant.

I HOPE ITS HELP YOU DEAR,
THANKS