Question

evaluate the integral. (remember to use absolute values where appropriate. use c for the constant of integration.) 6x sec(x) tan(x) dx

Answer 1

Integrate by parts.

`\displaystyle \int 6x \sec(x) \tan(x) \, dx = uv - \int v\,du`

with

`u = 6x \implies du = 6\,dx`

`dv = \sec(x)\tan(x)\,dx \implies v = \sec(x)`

so that

`\displaystyle \int 6x \sec(x) \tan(x) \, dx = 6x\sec(x) - 6 \int \sec(x) \, dx \\\\ ~~~~~~~~ = \boxed\sec(x) + \tan(x)`

where the last integral follows from

`\displaystyle \int \sec(x) \, dx = \int (\sec(x)(\sec(x)+\tan(x)))/(\sec(x)+\tan(x)) \, dx = \int (d(\sec(x)+\tan(x)))/(\sec(x)+\tan(x))`