Question

question 5
evaluate the integral by making the given substitution.
\\(\int \sec(4x)\tan(4x) dx,\quad u = 4x\\)
\\(\boxed{\quad} + c\\)
question help: \\(\text{video}\\) \\(\text{message instructor}\\)
submit question jump to answer

Explanation:

Step1: Define substitution and find $du$

Let $u = 4x$. Differentiate both sides:
$$\frac{du}{dx} = 4 \implies dx = \frac{1}{4}du$$

Step2: Substitute into integral

Replace $4x$ with $u$ and $dx$ with $\frac{1}{4}du$:
$$\int \sec(u)\tan(u) \cdot \frac{1}{4}du$$

Step3: Factor out constant and integrate

Factor out $\frac{1}{4}$, use $\int \sec(u)\tan(u)du = \sec(u) + C$:
$$\frac{1}{4}\int \sec(u)\tan(u)du = \frac{1}{4}\sec(u) + C$$

Step4: Substitute back $u=4x$

Replace $u$ with the original expression:
$$\frac{1}{4}\sec(4x)$$

Answer:

$\frac{1}{4}\sec(4x)$