Question

How do you integrate `int sec^2xtanx`?

Answer 1

You have to remeber that `sec^2(x)` is the derivative of `tan(x)`

Explanation:

So:

`int sec^2 x tan x dx = int tan x d (tan x) = (tan^2 x ) / 2`

Answer 2

`sec^2x/2+C` or `tan^2x/2+C`

Explanation:

You can also see this this way:

`intsec^2xtanxdx=intsecx(secxtanx)dx`

If `u=secx` then `du=secxtanxdx` so

`=intudu=u^2/2=sec^2x/2+C`

This is equivalent to the other answer of `tan^2x/2+C` because `tan^2x` and `sec^2x` are only a constant away from one another through the equality `tan^2x+1=sec^2x`.