dx-cscx-1-

Question Number 187526 by sciencestudentW last updated on 18/Feb/23

\int \frac{d x}{c s c x - 1} = ?

Answered by ARUNG_Brandon_MBU last updated on 18/Feb/23

I = \int \frac{d x}{cosec x - 1}

= \int \frac{\sin x}{1 - \sin x} d x = \int (\frac{1}{1 - \sin x} - 1) d x

= \int \frac{1 + \sin x}{\cos^{2} x} d x - x = \tan x + \frac{1}{\cos x} - x + C

Answered by floor(10²Eta[1]) last updated on 18/Feb/23

\int \frac{senx}{1 - senx} dx

u = tg (\frac{x}{2}) \Rightarrow \frac{2}{u^{2} + 1} du = dx

\int \frac{4 u}{{(u - 1)}^{2} (u^{2} + 1)} du = \int \frac{2}{{(u - 1)}^{2}} du - \int \frac{2}{u^{2} + 1} du

= - \frac{2}{u - 1} - 2 arctg (u) + C

= \frac{2}{1 - tg (\frac{x}{2})} - x + C