$$\int\sqrt{\smash[b]{\sec(x)}}\ln(\sec(x))\tan(x)\,\mathrm{d}x$$
I started by making u-substitution $u = \sec(x)$:
$$\int\sqrt u\ln(u)\tan(x) \left(\frac{\mathrm{d}u}{\sec(x)\tan(x)}\right)$$
Now, does the $\tan(x)$ cancel? Then is integration by parts the appropriate method to use?
Best Answer
Yes! The $\tan(x)$ cancels.
I'm sure you know what to do with the remaining $\sec(x)$, given that $u=\sec(x)$.
Integration by parts is next.