Question

`int (cos x)/(1 - sin x) "dx" =` ______.

पर्याय

• log |sec x + tan x| + c

• log |sec x(sex x + tan x)| + c

• log |sex - tan x| + c

• log |sec x(sec x - tan x)| + c

Answer 1

`int (cos x)/(1 - sin x) "dx" =` log |sec x(sex x + tan x)| + c.

Explanation:

Let I = `int (cos x)/(1 - sin x) "dx"`

Put 1 - sin x = t ⇒ cos x dx = - dt

`therefore "I" = int(- "dt")/"t" = - log |"t"|` + c

= - log |1 - sin x| + c

= log |(1 - sin x)^-1| + c

`= log |1/(1 - sin x)|` + c

`= log |1/(1 - sin x) xx (1 + sin x)/(1 + sin x)|`+ c

`= log |(1 + sin x)/(1 - sin ^2x)|`+ c

`= log |(1 + sin x)/cos^2x|` + c

`= log |1/(cos^2x) + (sin x)/(cos^2 x)|` + c

= log |sec² x + sec x · tan x| + c

= log |sec x (sec x + tan x)| + c