Question

cosx/cosx-1,Integrate the given function w.r.t. x considering them well defined and integrable over proper domain.

Answer 1

In the attachments I have answered

this problem.

I have applied decomposition method to find the anti derivative of the given function.

In the attachment I have answered this problem.

Answer 2

Dear Student,

Answer: ∫ cosx/cosx-1 dx=
$x + cot( \frac{x}{2} ) + c$

Solution:

$\int{\frac{ cos \: x}{ (cos \: x- 1 )}} dx$

$\int{\frac{cosx + 1 - 1}{cos \: x - 1}}$

split the term like

$\int{\frac{cos \: x - 1}{cos \: x - 1}}dx +\int{\frac{1}{cos \: x - 1}} dx\\ \\ = \int {1}dx \: -\int{ \frac{1}{1 -cos \: x }} dx$

as we know that

$cos \: 2x = 1 - 2 {sin}^{2} x$

$= \int{1}dx \: - \int{\frac{1}{2 {sin}^{2} ( \frac{x}{2}) } }dx \\ = \int{1}dx \: - \int{\frac{1}{2 } {cosec}^{2} (\frac{x}{2} ) }dx$

as we know that ∫$1dx = x + c$

∫${cosec}^{2} dx = - cot \: x + c$

so apply formula

$= x + \frac{1}{2} \times ( \frac{1}{ \frac{1}{2} } )cot( \frac{x}{2} ) + c \\ \\ = x + cot (\frac{x}{2} ) + c$

Hope it helps you.