Question

Find the integration of dt/(1-t)t​

Answer 1

$\int{ \frac{1}{t(1 - t)} }.dt$

$= \int{( \frac{1}{t} + \frac{1}{1 - t}) }.dt$

$= \int{ \frac{1}{t} } + \int{ \frac{1}{ 1 - t} }$

$= log(t) + \int{ \frac{1}{ 1 - t} }$

Now, let (1-t) = m

=> -dt = dm

Thus, Integral becomes:

$= log(t) + \int{ \frac{ 1}{m}.( - dm) }$

$= log(t) - \int{ \frac{ 1}{m}.dm}$

$= log(t) - log(m) + c$

Bring 'm=1-t' back,

= log(t) - log(1-t) + c

$= log( \frac{t}{1 - t} ) + c$

:)