Question

Solve each of the following integral:

\[\int_2^4 \frac{x}{x^2 + 1}dx\]

Answer 1

\[\int_2^4 \frac{x}{x^2 + 1}dx\]
\[ = \frac{1}{2} \int_2^4 \frac{2x}{x^2 + 1}dx\]
\[ = \frac{1}{2} \times \left.\log\left( x^2 + 1 \right)\right|_2^4 ...................\left[ \int\frac{f'\left( x \right)}{f\left( x \right)}dx = \log f\left( x \right) + C \right]\]
\[ = \frac{1}{2}\left( \log17 - \log5 \right)\]
\[ = \frac{1}{2}\log\left( \frac{17}{5} \right) .............\left( \log a - \log b = \log\frac{a}{b} \right)\]