Word Count : 521

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We are asked to derive a reduction formula for [ I_{m,n} = \int x^m (\log x)^n , dx ] and then use it to compute (\int x^4 (\log x)^3 , dx). --- Step 1: Derive the reduction formula We use integration by parts: Let [ u = (\log x)^n \quad \Rightarrow \quad du = n (\log x)^{n-1} \frac{dx}{x} = \frac{n (\log x)^{n-1}}{x} dx ] [ dv = x^m dx \quad \Rightarrow \quad v = \frac{x^{m+1}}{m+1}, \quad m \neq -1 ] Integration by parts formula: [ \int u , dv = uv - \int v , du ] Apply: [ I_{m,n} = \int x^m (\log ______ _______ _________ _____ ___ _________ ___ _______ _______ ________ ______.
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