Question

evaluate this integral \\(\int -4\tan x dx\\)

Explanation:

Step1: Factor out the constant

We know that for an integral \(\int kf(x)dx\), where \(k\) is a constant, we can factor out the constant: \(\int -4\tan x dx=-4\int \tan x dx\)

Step2: Recall the integral of \(\tan x\)

The integral of \(\tan x=\frac{\sin x}{\cos x}\) can be found using substitution. Let \(u = \cos x\), then \(du=-\sin xdx\), and \(\int\frac{\sin x}{\cos x}dx=-\int\frac{du}{u}=-\ln|u| + C=-\ln|\cos x|+C\)
So, \(-4\int\tan x dx=-4(-\ln|\cos x|)+C = 4\ln|\cos x|+C\)

Answer:

\(4\ln|\cos x| + C\) (where \(C\) is the constant of integration)