Show that the integral of tan(x) is ln|sec(x)| + C where C is a constant.

First, recall that tan(x) can be rewritten in terms of sine and cosine.

tan(x) = sin(x)/cos(x)

The rephrasing of our question suggests that we should try the substitution rule of integration.

We should substitute u=cos(x), since then du = -sin(x) dx and so sin(x) dx = -du

So the integral of tan(x) = the integral of sin(x)/cos(x) = the integral of -1/u = - ln|u| +C = - ln|cosx| +C

Now, - ln|cos(x)| = ln(|cos(x)|-1) = ln(1/|cos(x)|) = ln|sec(x)|

Therefore, the integral of tan(x) is ln|sec(x)| + C

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