Evaluate the following integrals: ^ n . dx - Vidyadip

Question

Evaluate the following integrals:$\int\text{x}^{\text{n}}.\log\text{x dx}$

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Answer

$\int\text{x}^{\text{n}}\log\text{x dx}$
Taking log x as the first function and $x^n$ as the second function.
$=\log\text{x}\int\text{x}^\text{n}\text{dx}-\int\Big(\frac{\text{d}}{\text{dx}}\log\text{x}\int\text{x}^\text{n}\text{dx}\Big)\text{dx}$
$=\log\text{x}\bigg(\frac{\text{x}^{\text{n}+1}}{\text{n}+1}\bigg)-\int\frac{1}{\text{x}}\bigg(\frac{\text{x}^{n+1}}{\text{n}+1}\bigg)\text{dx}$
$=\log\text{x}\bigg(\frac{\text{x}^{\text{n}+1}}{\text{n}+1}\bigg)-\int\frac{\text{x}^{\text{n}}}{\text{n}+1}\text{dx}$
$=\log\text{x}\bigg(\frac{\text{x}^{\text{n}+1}}{\text{n}+1}\bigg)-\int\frac{\text{x}^{\text{n}+1}}{(\text{n}+1)^2}+\text{C}$

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