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Proof: Integral csc(x)
(Math | Calculus | Integrals | Table Of | csc x)
 
Discussion of
(integral) csc x = - ln|csc x + cot x| + C.

1. Proof

    Strategy: The strategy is not obvious.  Multiply and divide by (csc x + cot x); use Substitution.
     
    (integral) csc x dx = (integral) csc x  csc x + cot x 
    csc x + cot x
    dx
    set
      u = csc x + cot x
    then we find
      du = (- csc x cot x - csc2 x) dx

    substitute du = (- csc x cot x - csc2 x) dx, u = csc x + cot x
     
    (integral) csc x  csc x + cot x 
    csc x + cot x
    dx = -(integral)
    (- csc2 x - csc x cot x) dx 
    csc x + cot x
     
    = -(integral)  du
    u

    solve integral

    = - ln |u| + C

    substitute back u=csc x + cot x

    = - ln |csc x + cot x| + C
    Q.E.D.

  
 
  

 
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