Integrate sec (4x) tan (4x) dx

sec (4x) + C

Explanation:

original problem: ∫sec (4x) tan (x) dx

use integration by substitution (u-sub) by setting u = 4x

if u = 4x, then du/dx = 4 and du = 4dx (dx = du/4)

after substitution the integral is ∫sec (u) tan (u) (du/4)

move the 1/4 out of the integral by using the integral Constant rule to form 1/4∫sec (u) tan (u) du

the anti-derivative of sec (u) tan (u) is sec (u), memorize your trigonometric derivatives!

after integration, we get sec (u) / 4 + C, now plug u back into the equation

sec (4x) + C is the general solution