Question

Calculate the following integrals.
${\int }_0^{2e}\frac{dx}{0.5x+1}.$

Answer 1

${\int }_0^{2e}\frac{dx}{0.5x+1}$

$\Rightarrow {\int }_0^{2e}\frac{dx}{\left(0.5\right)[x+\frac{1}{0.5}]}={\int }_0^{2e}\frac{dx}{\left(0.5\right)[x+2]}$

$=\frac{1}{0.5}\left(\log \right(x+2\left){|}_0^{2e}\right)$

$=\frac{1}{0.5}\left(\log \right(2e)-\log 2)$

$=\frac{1}{0.5}(\log 2+\log e-\log 2)$

$=\frac{1}{0.5}\left(\log e\right)$

$=\frac{1}{0.5}$.