Homework Help

How to integrate cosec x with respect to x?

user profile pic

stevesam93 | Student, Undergraduate | (Level 1) Honors

Posted October 18, 2012 at 10:13 AM via web

dislike 2 like

How to integrate cosec x with respect to x?

1 Answer | Add Yours

user profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted October 18, 2012 at 12:44 PM (Answer #1)

dislike 2 like

The integral `int cosec x dx` has to be determined.

 `int cosec x dx`

=> `int 1/sin x dx`

=> `int sin x/(sin^2x) dx`

=> `int sin x/(1 - cos^2x) dx`

let `y = cos x => dy = -sin x dx`

=> `int 1/(y^2 - 1) dy`

=> `(1/2)*int ((y + 1) - (y - 1))/(y^2 - 1) dy`

=> `(1/2)*int 1/(y - 1) dy - (1/2)*int 1/(y + 1) dy`

=> `(ln(y -1) - ln(y + 1))/2`

substitute y = cos x

=> `(ln(cos x - 1) - ln(cos x + 1))/2 + C`

The integral `int cosec x dx = (ln(cos x - 1) - ln(cos x + 1))/2 + C`

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes