By multiplying by a form of 1 evaluate 1/cscx-cotx dx

Question

Its under trigonometric functions in calculus under derivatives

Answer 1

 

1/(cosec(x)-cot(x))dx=sin(x)dx/(1-cos(x)), by multiplying top and bottom by sin(x).

INTEGRATION

Let y=1-cos(x) so y'=sin(x) and dy=sin(x)dx so to evaluate ∫(sin(x)/(1-cos(x))dx we can replace sin(x)dx with dy and we have ∫dy/y=ln|y|=ln|1-cos(x)| or more generally ln|A(1-cos(x)| where A is a constant.