# Homework Help: Integral of tan(x)^3

1. Mar 6, 2008

### Goldenwind

[SOLVED] Integral of tan(x)^3

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

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

Book says I'm wrong. Where is my mistake?

Last edited: Mar 6, 2008
2. Mar 6, 2008

### Mathdope

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.

3. Mar 6, 2008

### Goldenwind

...wow.... damnit.
lol, thank-you.

4. Oct 22, 2010

### shelovesmath

Re: [SOLVED] Integral of tan(x)^3

I'm wondering if this method is also legal.

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5. Oct 22, 2010

### Quinzio

Re: [SOLVED] Integral of tan(x)^3

yes sure

6. Oct 22, 2010

### shelovesmath

Re: [SOLVED] Integral of tan(x)^3

Ok, but my answer is different. . .