(1)/( \sec(1 - \sin(x) ) ) dx Any help with this? ​

Question

`(1)/( \sec(1 - \sin(x) ) ) dx`
Any help with this? ​

Answer 1

`\displaystyle(1)/(\sec x(1-\sin x))=(\cos x)/(1-\sin x)\\\\\\\int (dx)/(\sec x(1-\sin x))=\int(\cos x\, dx)/(1-\sin x)\\\\\\u=1-\sin x\\du=-\cos x\, dx\\\\\int (\cos x\, dx)/(1-\sin x)=\int (-du )/(u)=-\int (du)/(u)=-\ln |u|+C=-\ln|1-\sin x|+C`