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Question
\[\int \log_{10} x\ dx\]
Sum
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Solution
\[\int \log_{10} x\ dx\]
\[ = \int\frac{\log_e x}{\log_e 10} dx\]
\[ = \frac{1}{\log_e 10}\int 1_{II} \cdot \log_I x \text{ dx}\]
\[ = \frac{1}{\log_e 10}\left[ \log_e x\int1 \text{ dx} - \int\left\{ \frac{d}{dx}\left( \log_e x \right)\int1 \text{ dx} \right\}\text{ dx}\right]\]
\[ = \frac{1}{\log_e 10}\left[ \log_e x \cdot x - \int\frac{1}{x} \times x \text{ dx} \right]\]
\[ = \frac{1}{\log_e 10}\left[ x \log_e x - x \right] + C\]
\[ = \frac{1}{\log_e 10} \times x \left( \log_e x - 1 \right) + C\]
\[ = x \left( \log_e x - 1 \right) \cdot \log_{10} e + C\]
\[ = \int\frac{\log_e x}{\log_e 10} dx\]
\[ = \frac{1}{\log_e 10}\int 1_{II} \cdot \log_I x \text{ dx}\]
\[ = \frac{1}{\log_e 10}\left[ \log_e x\int1 \text{ dx} - \int\left\{ \frac{d}{dx}\left( \log_e x \right)\int1 \text{ dx} \right\}\text{ dx}\right]\]
\[ = \frac{1}{\log_e 10}\left[ \log_e x \cdot x - \int\frac{1}{x} \times x \text{ dx} \right]\]
\[ = \frac{1}{\log_e 10}\left[ x \log_e x - x \right] + C\]
\[ = \frac{1}{\log_e 10} \times x \left( \log_e x - 1 \right) + C\]
\[ = x \left( \log_e x - 1 \right) \cdot \log_{10} e + C\]
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