Question

evaluate the integral by making the given substitution. \\( \int \sec(4x) \tan(4x) dx, \quad u = 4x \\) \\( \boxed{} + c \\)

Explanation:

Step1: Recall substitution rule

Given \( u = 4x \), first find \( du \). Differentiating \( u \) with respect to \( x \), we get \( du=\frac{d}{dx}(4x)dx = 4dx \), so \( dx=\frac{du}{4} \).

Step2: Substitute into integral

The integral is \( \int\sec(4x)\tan(4x)dx \). Substitute \( u = 4x \) and \( dx=\frac{du}{4} \), we have:
\[

$$\begin{align*} \int\sec(u)\tan(u)\cdot\frac{du}{4}&=\frac{1}{4}\int\sec(u)\tan(u)du \end{align*}$$

\]

Step3: Integrate \( \sec(u)\tan(u) \)

We know that the antiderivative of \( \sec(u)\tan(u) \) with respect to \( u \) is \( \sec(u)+C \) (from integral formulas: \( \int\sec(u)\tan(u)du=\sec(u)+C \)). So,
\[
\frac{1}{4}\int\sec(u)\tan(u)du=\frac{1}{4}\sec(u)+C
\]

Step4: Substitute back \( u = 4x \)

Replace \( u \) with \( 4x \), we get \( \frac{1}{4}\sec(4x)+C \).

Answer:

\(\frac{1}{4}\sec(4x)\) (the \( +C \) is already accounted for in the problem's request for the integral result with \( +C \))