Question
If , show that:
Hence find the value of .
Answer :
Word Count : 521
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Skip in 10
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 x)^n dx = \frac{x^{m+1}}{m+1} (\log x)^n - \int \frac{x^{m+1}}{m+1} \cdot \frac{n (\log x)^{n-1}}{x} dx ] Simplify the integral: [ \int \frac{x^{m+1}}{m+1} \cdot \frac{n (\log x)^{n-1}}{x} dx = \frac{n}{m+1} \int x^m (\log x)^{n-1} dx = \frac{n}{m+1} I_{m,n-1} ]
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Related Questions
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समीकरण हल कीजिए, जिसके सभी मूल G.P. में हैं।
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चक्रज की लम्बाई ज्ञात कीजिए और दर्शाइए कि रेखा
इसे 1 : 3 के अनुपात में विभक्त करती है।
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Is () associative in ? Is (.) distributive over (*) in
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