Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ 4x sec²(2x) dx

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Identify the parts of the integral for integration by parts. Recall the formula: $\int u \, dv = uv - \int v \, du$. Choose $u$ and $dv$ from the integral $\int 4x \sec^{2}(2x) \, dx$.

Let $u = 4x$ because its derivative simplifies the expression, and let $dv = \sec^{2}(2x) \, dx$ because it is straightforward to integrate.

Compute $du$ by differentiating $u$: $du = 4 \, dx$. Next, find $v$ by integrating $dv$: $v = \int \sec^{2}(2x) \, dx$. Remember to use substitution for the inner function \$2x$.

Apply the integration by parts formula: $\int 4x \sec^{2}(2x) \, dx = uv - \int v \, du$. Substitute the expressions for $u$, $v$, and $du$ into this formula.

Simplify the resulting integral and evaluate it. This may involve another substitution or direct integration. Finally, add the constant of integration $C$ to your answer.

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