Question
If , show that:
Hence find the value of .
Answer :
Word Count : 96
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 _______ _______ ______ ________ __________.
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___ ______ ________ ______ ______ __________ ______ _________ ____.
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________ _________ ______ ________ _____ _____ _______.
________ ___ _________ ________ ____ ______ _____ _______.
_________ _________ ________ ________ __________ ___ _______.
____ __________ _______ ________.
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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 _______ _______ ______ ________ __________.
________ _____ ____ ________ _______ _____ _______.
____ ________ ________ _________ _________.
____ _____ _______ _____ ______ _________ _____ ___.
___ ______ ________ ______ ______ __________ ______ _________ ____.
________ __________ __________ _____ __________ ______.
_______ ___ ________ ________ _______ ________ __________ __________ _______ _____ ___.
________ _________ ______ ________ _____ _____ _______.
________ ___ _________ ________ ____ ______ _____ _______.
_________ _________ ________ ________ __________ ___ _______.
____ __________ _______ ________.
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