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Integration Calculus

Integrate the following with respect to x:

sec(x) / cos(2x)

Try to simplify it using one of the following identities:

$cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x)$

$sec(x) = \frac 1{cos(x)}$

then try to integrate the resultant.

$\int\frac{sec(x)}{cos(2x)}dx$

This one is fairly difficult and not easily done by elementary means.

As I said previously, this won't be pretty.

$\int\frac{sec(x)}{cos(2x)}dx=\int sec(x)sec(2x)dx$

One substitution we could make is to let $x=2tan^{-1}(u).\;\ dx=\frac{2}{u^{2}+1}du$

Making the subs and converting to partial fractions gives us:

$-2\int\frac{1}{u^{2}+2u-1}du-2\int\frac{1}{u^{2}-2u-1}du-\int\frac{1}{u+1}du+\int\frac{1}{u+1}du$

Integrate term by term and resub.

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