Word Count : 653

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यहाँ हमें दिखाना है कि: \[ I_{m,n} = \int x^3 (\log x)^n \, dx \] और फ़ंक्शन: \[ f(x) = 2x^2 - 7x - 10 \quad \text{over} \quad [2,5] \] हालांकि, प्रश्न में \( I_{m,n} \) और \( f(x) \) के बीच सीधा संबंध नहीं दिया गया है, इसलिए हम यह मानेंगे कि हमें \( I_{m,n} \) को हल करना है और फिर \( f(x) \) के साथ किसी तरह का संबंध स्थापित करना है। ### चरण 1: \( I_{m,n} \) को हल करें हमें निम्नलिखित इंटीग्रल को हल करना है: \[ I_{m,n} = \int x^3 (\log x)^n \, dx \] इस इंटीग्रल को हल करने के लिए, हम इंटीग्रेशन बाय पार्ट्स (Integration by Parts) का उपयोग करेंगे। इंटीग्रेशन बाय पार्ट्स का सूत्र है: \[ \int u \, dv = ____ _________ ________ ________ ___ _________.
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