Word Count : 393

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हम दिए गए इंटीग्रल को हल करने के लिए पार्ट्स द्वारा समाकलन (Integration by Parts) का उपयोग करेंगे। ### 1. सिद्ध करना: \((m+1)I_{m,n} = x^{m+1}(\log x)^n - nI_{m,n-1}\) दिया गया इंटीग्रल: \[ I_{m,n} = \int x^m (\log x)^n \,dx \] इसे हल करने के लिए हम Integration by Parts का उपयोग करेंगे, जहाँ हम निम्न प्रकार चुनते हैं: - पहला भाग (u): \((\log x)^n\), जिससे \(du = n(\log x)^{n-1} \cdot \frac{dx}{x}\) - दूसरा भाग (dv): \(x^m dx\), जिससे _________ _____ _________ _____ _________ ___ ___ ______ __________ _______ ________ ___.
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