Themes > Science > Mathematics > Calculus > Integrals > Integrals Table > Integral tan(x)
Discussion of (integral) tan x = - ln|cos x| + C.


1. Proof

    Strategy: Make in terms of sin's and cos's; Use Subtitution.
     
    (integral) tan x dx = (integral) sin x

    cos x

    		 dx
    set
      u = cos x.
    then we find
      du = - sin x dx

    substitute du=-sin x, u=cos x
     
    (integral) sin x

    cos x

    		 dx = - (integral)
    (-1) sin x dx

    cos x

     
    = - (integral) du

    u

    Solve the integral

    = - ln |u| + C

    substitute back u=cos x

    = - ln |cos x| + C
    Q.E.D.

2. Alternate Form of Result

    (integral) tan x dx = - ln |cos x| + C
    = ln | (cos x)-1 | + C
    = ln |sec x| + C

    Therefore:
    (integral) tan x dx = - ln |cos x| + C = ln |sec x| + C


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