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

$\displaystyle \int 6x \sec(x) \tan(x) \, dx = 6x\sec(x) - 6 \int \sec(x) \, dx \\\\ ~~~~~~~~ = \boxed{6x\sec(x) - 6 \ln|\sec(x) + \tan(x)| + C}$

where the last integral follows from

$\displaystyle \int \sec(x) \, dx = \int \frac{\sec(x)(\sec(x)+\tan(x))}{\sec(x)+\tan(x)} \, dx = \int \frac{d(\sec(x)+\tan(x))}{\sec(x)+\tan(x)}$