Answer
$$\int 4x\sec^22xdx=2x\tan2x+\ln|\cos2x|+C$$
Work Step by Step
$$A=\int 4x\sec^22xdx$$
Set $u=4x$ and $dv=\sec^22xdx$
Then we have $du=4dx$ and $v=\frac{1}{2}\tan2x$
Using the formula $\int udv= uv-\int vdu$:
$$A=4x\times\frac{1}{2}\tan2x-\int\frac{1}{2}\tan2x\times4dx$$ $$A=2x\tan2x-2\int\tan2xdx$$ $$A=2x\tan2x-2\int\frac{\sin2x}{\cos2x}dx$$
Take $a=\cos2x$, then $$da=-2\sin2xdx$$
That means $$\sin2xdx=-\frac{1}{2}da$$
Therefore, $$A=2x\tan2x-2\Big(-\frac{1}{2}\Big)\int\frac{da}{a}$$ $$A=2x\tan2x+1(\ln|a|)+C$$ $$A=2x\tan2x+\ln|\cos2x|+C$$
