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Integral of tan(x)^3

  1. Mar 6, 2008 #1
    [SOLVED] Integral of tan(x)^3

    1. The problem statement, all variables and given/known data
    [tex]\int tan^3 x dx[/tex]

    3. The attempt at a solution
    [tex]= \int tanx*tan^2x dx[/tex]
    [tex]= \int tanx*(1 - sec^2x) dx[/tex]
    [tex]= \int tanx - tanxsec^2x dx[/tex]
    [tex]= \int tanx dx - \int tanxsec^2x dx[/tex]
    [tex]= \int \frac{sinx}{cosx} dx - \int u du[/tex]
    [tex]= -\int \frac{-sinx}{cosx} dx - \frac{1}{2}u^2[/tex]
    [tex]= -\int \frac{D(cosx)}{cosx} dx - \frac{1}{2}tanx^2[/tex]
    [tex]= -ln(cosx) - \frac{1}{2}tanx^2 + C[/tex]
    [tex]= ln(cosx^{-1}) - \frac{1}{2}tanx^2 + C[/tex]
    [tex]= ln(\frac{1}{cosx}) - \frac{1}{2}tanx^2 + C[/tex]
    [tex]= ln(secx) - \frac{1}{2}tanx^2 + C[/tex]

    Book says I'm wrong. Where is my mistake?
    Last edited: Mar 6, 2008
  2. jcsd
  3. Mar 6, 2008 #2
    1 + tan^2 = sec^2 is not equivalent to 1 - sec^2 = tan^2? (step 2) Looks like you missed a negative sign, pretty small error that apparently got magnified later on.
  4. Mar 6, 2008 #3
    ...wow.... damnit.
    lol, thank-you.
  5. Oct 22, 2010 #4
    Re: [SOLVED] Integral of tan(x)^3

    I'm wondering if this method is also legal.

    Attached Files:

  6. Oct 22, 2010 #5
    Re: [SOLVED] Integral of tan(x)^3

    yes sure
  7. Oct 22, 2010 #6
    Re: [SOLVED] Integral of tan(x)^3

    Ok, but my answer is different. . .
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