# Math Help - integrate with trig functions

1. ## integrate with trig functions

integrate:

(1 - tan x) / (sec x)^2 dx

i have tried substitutions and used trig identies but cannot seem to figure this out

2. Split the fraction into two and simplify each part into terms of $\sin x$ and $\cos x$. So:

\begin{aligned} \int \frac{1 - \tan x}{\sec^2 x} \ dx & = \int \left(\frac{1}{\sec^2 x} - \frac{\tan x}{\sec^2 x}\right)dx \\ & = \int \left( \cos^2 x - \sin x \cos x \right) dx \end{aligned}

Now note that:
• $\cos^2 x = \frac{1 + \cos 2x}{2}$
• $\sin x \cos x = \tfrac{1}{2}\sin (2x)$