50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:
53. ∫ lnⁿ(x) dx = x lnⁿ(x) - n ∫ lnⁿ⁻¹(x) dx

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Identify the integral to solve: \(\int \ln^{n}(x) \, dx\), where \(n\) is a positive integer power of the natural logarithm function.

Choose parts for integration by parts. Let \(u = \ln^{n}(x)\) and \(dv = dx\). Then, compute \(du\) and \(v\): - Since \(u = \ln^{n}(x)\), use the chain rule to find \(du = n \ln^{n-1}(x) \cdot \frac{1}{x} \, dx\). - Since \(dv = dx\), integrate to get \(v = x\).

Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Substitute the expressions for \(u\), \(v\), and \(du\): \(\int \ln^{n}(x) \, dx = x \ln^{n}(x) - \int x \cdot n \ln^{n-1}(x) \cdot \frac{1}{x} \, dx\).

Simplify the integral inside the subtraction: \(\int x \cdot n \ln^{n-1}(x) \cdot \frac{1}{x} \, dx = n \int \ln^{n-1}(x) \, dx\).

Rewrite the expression to obtain the reduction formula: \(\int \ln^{n}(x) \, dx = x \ln^{n}(x) - n \int \ln^{n-1}(x) \, dx\).

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주요 개념

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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. This method is essential for deriving reduction formulas involving powers of logarithmic functions.

Reduction Formulas

Reduction formulas express an integral involving a power or parameter in terms of a similar integral with a lower power or simpler parameter. They simplify complex integrals step-by-step, making it easier to evaluate integrals like ∫lnⁿ(x) dx by relating them to ∫lnⁿ⁻¹(x) dx.

Properties of the Natural Logarithm Function

Understanding the natural logarithm function ln(x) and its derivatives is crucial. Since d/dx[ln(x)] = 1/x, this property helps in choosing appropriate parts for integration by parts and manipulating integrals involving powers of ln(x).

50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:53. ∫ lnⁿ(x) dx = x lnⁿ(x) - n ∫ lnⁿ⁻¹(x) dx