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