Imin=∫xm(logx)ndx =xm+1m+1(logx)n−nm+1∫(logx)n−1. xm+1.x−1dx =xm+1m+1(logx)n−nm+1Im,n−1

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$I_{m, n} = \int x^{m} {(\log x)}^{n} d x =$

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$\frac{x^{m + 1}}{m + 1} {(\log x)}^{n} - \frac{n}{m + 1} I_{m, n - 1}$

$\frac{x^{m}}{m} {(\log x)}^{n} - \frac{n - 1}{m - 1} I_{m - 1, n - 1}$

$\frac{x^{m + 1}}{m} \frac{{(\log x)}^{n H}}{n + 1} - \frac{n}{n - 1} I_{m, n - 1}$

$\frac{x^{m} {(\log x)}^{n + 1}}{n + 1} - \frac{n}{m + 1} I_{m, n - 1}$

answer is A.

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Detailed Solution

$\begin{array}{l} I_{\min} = \int x^{m} {(\log x)}^{n} d x \\ = \frac{x^{m + 1}}{m + 1} {(\log x)}^{n} - \frac{n}{m + 1} \int {(\log x)}^{n - 1} . x^{m + 1} . x^{- 1} d x \\ = \frac{x^{m + 1}}{m + 1} {(\log x)}^{n} - \frac{n}{m + 1} I_{m, n - 1} \end{array}$