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If $\int(\log x)^3 x^5 d x=\frac{x^6}{A}\left[B(\log x)^3+C(\log x)^2+\right.$ $\mathrm{D}(\operatorname{lug} \mathrm{x})-1]+\mathrm{k}$ and $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ are integers, then $A-(B+C+D)=$
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$192$
$\int(\log x)^3 x_{\text {II }}^5 d x$
$\begin{aligned} & =(\log x)^3 \cdot \int x^5 d x-\int\left(\frac{d}{d x}(\log x)^3\right) \cdot\left(\int x^5 d x\right) d x \\ & =\frac{x^6}{6}(\log x)^3-\int\left(\frac{3(\log x)^2}{x} \cdot \frac{x^6}{6}\right) d x\end{aligned}$
$\begin{aligned} & =\frac{x^6}{6}(\log x)^3-\frac{1}{2} \int(\log x)^2 x^5 d x \\ & =\frac{x^6}{6}(\log x)^3-\frac{1}{2}\left[(\log x)^2 \cdot \frac{x^6}{6}-\int 2 \frac{\log x}{x} \cdot \frac{x^6}{6} d x\right] \\ & =\frac{x^6}{6}(\log x)^3-\frac{1}{2}\left[\frac{x^6}{6}(\log x)^2-\frac{1}{3} \int \log x \cdot x_{\text {II }}^5 d x\right] \\ & =\frac{x^6}{6}\left[(\log x)^3-\frac{1}{2}(\log x)^2\right]+\frac{1}{6}\left[\frac{x^6}{6} \cdot \log x-\int \frac{1}{x} \cdot \frac{x^6}{6} d x\right] \\ & =\frac{x^6}{6}\left[(\log x)^3-\frac{1}{2}(\log x)^2+\frac{1}{6} \log x\right]-\frac{1}{6 \times 6} \int x^5 d x \\ & =\frac{x^6}{6}\left[(\log x)^3-\frac{1}{2}(\log x)^2+\frac{1}{6} \log x-\frac{1}{36}\right]+k \\ & =\frac{x^6}{6}\left[\frac{36(\log x)^3-18(\log x)^2+6(\log x)-1}{36}\right]+k\end{aligned}$
$\begin{aligned} & =\frac{x^6}{216}\left[36(\log x)^3-18(\log x)^2+6(\log x)-1\right]+k \\ & A=216, B=36, C=-18, D=6 \\ & A-(B+C+D) \\ & =216-(36-18+6)=192\end{aligned}$
$\begin{aligned} & =(\log x)^3 \cdot \int x^5 d x-\int\left(\frac{d}{d x}(\log x)^3\right) \cdot\left(\int x^5 d x\right) d x \\ & =\frac{x^6}{6}(\log x)^3-\int\left(\frac{3(\log x)^2}{x} \cdot \frac{x^6}{6}\right) d x\end{aligned}$
$\begin{aligned} & =\frac{x^6}{6}(\log x)^3-\frac{1}{2} \int(\log x)^2 x^5 d x \\ & =\frac{x^6}{6}(\log x)^3-\frac{1}{2}\left[(\log x)^2 \cdot \frac{x^6}{6}-\int 2 \frac{\log x}{x} \cdot \frac{x^6}{6} d x\right] \\ & =\frac{x^6}{6}(\log x)^3-\frac{1}{2}\left[\frac{x^6}{6}(\log x)^2-\frac{1}{3} \int \log x \cdot x_{\text {II }}^5 d x\right] \\ & =\frac{x^6}{6}\left[(\log x)^3-\frac{1}{2}(\log x)^2\right]+\frac{1}{6}\left[\frac{x^6}{6} \cdot \log x-\int \frac{1}{x} \cdot \frac{x^6}{6} d x\right] \\ & =\frac{x^6}{6}\left[(\log x)^3-\frac{1}{2}(\log x)^2+\frac{1}{6} \log x\right]-\frac{1}{6 \times 6} \int x^5 d x \\ & =\frac{x^6}{6}\left[(\log x)^3-\frac{1}{2}(\log x)^2+\frac{1}{6} \log x-\frac{1}{36}\right]+k \\ & =\frac{x^6}{6}\left[\frac{36(\log x)^3-18(\log x)^2+6(\log x)-1}{36}\right]+k\end{aligned}$
$\begin{aligned} & =\frac{x^6}{216}\left[36(\log x)^3-18(\log x)^2+6(\log x)-1\right]+k \\ & A=216, B=36, C=-18, D=6 \\ & A-(B+C+D) \\ & =216-(36-18+6)=192\end{aligned}$
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