Question

\[\int \log_{10} x\ dx\]

Answer 1

\[\int \log_{10} x \text{ dx }\]
\[ = \int\frac{\log x}{\log 10}dx\]
\[ = \frac{1}{\log 10}\int 1_{} \cdot \text{ log x  dx }\]
`  " Taking log x as the first function and 1 as the second function " `
\[ = \frac{1}{\log 10}\left[ \log x \int\text{ 1 dx} - \int\left\{ \frac{d}{dx}\left( \log x \right)\int\text{ 1 dx }\right\}dx \right]\]

\[ = \frac{1}{\log 10}\left[ \log x \cdot x - \int\frac{1}{x} \cdot \text{ x dx } \right]\]
\[ = \frac{1}{\log 10}\left[ x \log x - x \right] + C\]
\[ = \frac{1}{\log 10}\left[ x\left( \log x - 1 \right) \right] + C\]