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

Verified step by step guidance

1. 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 \(\int 4x \sec^{2}(2x) \, dx\).
2. Let \(u = 4x\) because its derivative simplifies the expression, and let \(dv = \sec^{2}(2x) \, dx\) because it is straightforward to integrate.
3. 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$.
4. 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.
5. 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.

Key Concepts

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is crucial to simplify the problem.

Derivative and Integral of Trigonometric Functions

Understanding the derivatives and integrals of trigonometric functions like sec²(x) is essential. For example, the integral of sec²(x) is tan(x), and knowing these helps in evaluating integrals involving trigonometric expressions efficiently.

Substitution Method within Integration by Parts

Sometimes, after applying integration by parts, substitution is needed to handle inner functions, especially when the integrand includes composite functions like sec²(2x). Recognizing when and how to apply substitution simplifies the integral evaluation.