Question
If , show that:
Hence find the value of .
Answer :
Word Count : 96
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Let (I_{m,n}=\int x^{m}(\log x)^n,dx). Integrate by parts with (u=(\log x)^n), so (du=\dfrac{n(\log x)^{,n-1}}{x},dx), and (dv=x^m dx), so (v=\dfrac{x^{m+1}}{m+1}). Then [ I_{m,n}=\frac{x^{m+1}}{m+1}(\log x)^n-\frac{n}{m+1}\int x^m(\log x)^{n-1}dx. ] Hence, [ (m+1)I_{m,n}=x^{m+1}(\log x)^n-nI_{m,n-1}. ] Now take (m=4), (n=3). Then
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