Question

How do you find the antiderivative of `(sinx - cosx)/cosx dx`?

Answer 1

Hello,

The answer is `-ln(|cos(x)|) - x + c`.

Write `\frac{\sin(x)-\cos(x)}{\cos(x)} = \frac{\sin(x)}{\cos(x)} - 1 = -\frac{\cos'(x)}{cos(x)} -1`.

You recognize `\int (u')/u = ln(|u|) + c`, therefore the antiderivative of your function is

`-ln(|cos(x)|) - x + c`

where c is an arbitrary real constant.