Write a value of ( )^ n x dx - Vidyadip

Question

Write a value of $\int\frac{(\log\text{x})^{\text{n}}}{\text{x}}\text{ dx}$

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Answer

Let $\text{I}=\int\frac{(\log\text{x})^{\text{n}}}{\text{x}}\text{ dx}$
Let $\log\text{x}=\text{t}$
$\frac{1}{\text{x}}\text{ dx}=\text{dt}$
$\text{dx}=\text{xdt}$
$\therefore\ \int\frac{(\log\text{x})^{\text{n}}}{\text{x}}\text{ dx}=\int(\text{t})^{\text{n}}\text{dt}$
$=\frac{\text{t}^{\text{n}+1}}{\text{n}+1}+\text{C}$
$=\frac{(\log\text{x})^{\text{n}+1}}{(\text{n}+1)}+\text{C}$

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