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(D) To evaluate the integral $\int x^n \log x \, dx$,we use the method of integration by parts: $\int u \, dv = uv - \int v \, du$.Let $u = \log x$ an

Vedclass · Accepted Answer

$\frac{x^{n+1}}{n+1} \left\{ \log x - \frac{1}{n+1} \right\} + c$

Vedclass · Answer

(D) To evaluate the integral $\int x^n \log x \, dx$,we use the method of integration by parts: $\int u \, dv = uv - \int v \, du$.Let $u = \log x$ and $dv = x^n \, dx$.Then $du = \frac{1}{x} \, dx$ and $v = \frac{x^{n+1}}{n+1}$.Applying the formula:$\int x^n \log x \, dx = \log x \cdot \frac{x^{n+1}}{n+1} - \int \frac{x^{n+1}}{n+1} \cdot \frac{1}{x} \, dx$$= \frac{x^{n+1}}{n+1} \log x - \frac{1}{n+1} \int x^n \, dx$$= \frac{x^{n+1}}{n+1} \log x - \frac{1}{n+1} \cdot \frac{x^{n+1}}{n+1} + c$$= \frac{x^{n+1}}{n+1} \left( \log x - \frac{1}{n+1} \right) + c$.

$\int x^n \log x \, dx = $

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