Having trouble with the substitution method. I have to evaluate the integral:

I see that with u being equal to $4x$ I will get that $du/dx$ will equal $4$ and lead to $du=4dx$. However I am stuck when I plug in the u values to get $\sec u\cos u\,du$. The problem states to use the subsitution method but it looks like I will have to use the product rule because of $\sec u\cos u\,du$. Is there a way to evaluate this integral without the product rule? Thanks in advance

i think you can use the change of variable $u = \cos 4x, du = -4 \sin 4x \, dx$ instead.
then $$\int \sec 4x \tan 4x \, dx = \int \frac{\sin 4x}{\cos^2 4x} \, dx
\to -\frac 14\int \frac{du}{u^2} = \frac{1}{4u} = \frac 14\sec4x + C$$
