9–40. Integration by parts Evaluate the following integrals using integration by parts.
38. ∫ x² ln²(x) dx

Verified step by step guidance

1. Identify the integral to solve: \(\int x^{2} \ln^{2}(x) \, dx\).
2. Choose parts for integration by parts. Let \(u = \ln^{2}(x)\) (since its derivative simplifies the expression) and \(dv = x^{2} \, dx\) (which is straightforward to integrate).
3. Compute \(du\) and \(v\): - Differentiate \(u\): \(du = 2 \ln(x) \cdot \frac{1}{x} \, dx = \frac{2 \ln(x)}{x} \, dx\). - Integrate \(dv\): \(v = \int x^{2} \, dx = \frac{x^{3}}{3}\).
4. Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Substitute the expressions: \(\int x^{2} \ln^{2}(x) \, dx = \frac{x^{3}}{3} \ln^{2}(x) - \int \frac{x^{3}}{3} \cdot \frac{2 \ln(x)}{x} \, dx\).
5. Simplify the integral inside and prepare to solve \(\int \frac{2}{3} x^{2} \ln(x) \, dx\) using integration by parts again, following a similar process.

Key Concepts

Integration by Parts

Integration by parts is a technique derived from the product rule of 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 integral effectively.

Logarithmic Functions and Their Derivatives

Understanding the derivative of logarithmic functions, such as ln(x), is essential. The derivative of ln(x) is 1/x, and when dealing with powers like ln²(x), the chain rule applies. This knowledge helps in differentiating u when applying integration by parts.

Polynomial Functions and Their Integration

Polynomial functions like x² are straightforward to integrate and differentiate. Recognizing that the integral and derivative of polynomials are simpler helps in selecting dv or u in integration by parts, facilitating the reduction of the integral into manageable parts.