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To solve this problem, we'll begin by proving the given equation and then use it to find the value of the integral. ### Step 1: Prove the given equation The given integral is: \[ I_{m,n} = \int x^3 (\log x)^n \, dx \] We are tasked with showing that: \[ (m+1) I_{m,n} = x^{m+1} (\log x)^n - n I_{m,n-1} \] To do this, we apply integration by parts. Let's choose: - \( u = (\log x)^n \), so that \( du = n (\log x)^{n-1} \frac{1}{x} dx \). - \( dv = x^m dx \), so that \( v = \frac{x^{m+1}}{m+1} \). Now, applying the integration by parts formula \( \int u \, dv = uv - ________ ___ _____ ________ ____ ________.
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