## Step 1: Identify the Integral Form The given integral is: \[ \int \sec(8x)\tan(8x)\, dx \] Recall that the derivative of \(\sec(u)\) with respect to \(u\) is \(\sec(u)\tan(u)\). So if we let \(u = 8x\), then \(du = 8\,dx\). **Explanation:** We notice that \(\sec(8x)\tan(8x)\) is the derivative of \(\sec(8x)\), but we need to account for the chain rule since the argument is \(8x\). --- ## Step 2: Substitute and Integrate Let \(u = 8x\), so \(dx = \frac{du}{8}\): \[ \int \sec(8x)\tan(8x)\, dx = \int \sec(u)\tan(u) \cdot \frac{du}{8} \] \[ = \frac{1}{8} \int \sec(u)\tan(u)\, du \] Since \(\int \sec(u)\tan(u)\, du = \sec(u)\), we have: \[ = \frac{1}{8} \sec(u) + C \] \[ = \frac{1}{8} \sec(8x) + C \] **Explanation:** We performed a substitution and integrated, using the fact that the antiderivative of \(\sec(u)\tan(u)\) is \(\sec(u)\). --- ## **Final Answer** \[ \boxed{\int \sec(8x)\tan(8x)\, dx = \frac{1}{8} \sec(8x) + C} \]