`int 4/(csc(theta)-cot(theta)) d theta` Find or evaluate the integral

Given to solve ,

`int 4/(csc(theta)-cot(theta)) d theta`

just for the convenience let` x= theta`

so,` int 4/(csc(theta)-cot(theta)) d theta`

=`int 4/(csc(x)-cot(x)) d x`

=`int 4/((1/sin(x))-(cos(x)/sin(x))) d x`

=`int 4/((1-cosx)/sinx) dx`

= `int 4sinx/(1-cosx) dx`----------(1)

let `u = 1-cosx`

=> `du=sinx dx`

so , the equation (1) implies

`int 4sinx/(1-cosx) dx`

= `4 int (1/u) du`

= `4 ln(u)+c`

= `4ln(1-cosx)+c`

but `x= theta` , so

= `4ln(1-costheta)+c`

Comments

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