Answer
$x = \frac{{10}}{3}\left( {\ln 2} \right)$
Work Step by Step
$$\eqalign{
& \int_0^x {0.3{e^{ - 0.3t}}} dt = \frac{1}{2} \cr
& {\text{Integrating}} \cr
& 0.3\int_0^x {{e^{ - 0.3t}}} dt = \frac{1}{2} \cr
& 0.3\left[ {\frac{{{e^{ - 0.3t}}}}{{ - 0.3}}} \right]_0^x = \frac{1}{2} \cr
& - \left[ {{e^{ - 0.3t}}} \right]_0^x = \frac{1}{2} \cr
& {\text{Evaluating the limits of integration}} \cr
& - \left[ {{e^{ - 0.3x}} - {e^{ - 0.3\left( 0 \right)}}} \right] = \frac{1}{2} \cr
& - \left[ {{e^{ - 0.3x}} - 1} \right] = \frac{1}{2} \cr
& 1 - {e^{ - 0.3x}} = \frac{1}{2} \cr
& {\text{Solve the equation for }}x \cr
& {e^{ - 0.3x}} = 1 - \frac{1}{2} \cr
& {e^{ - 0.3x}} = \frac{1}{2} \cr
& \ln \left( {{e^{ - 0.3x}}} \right) = \ln \left( {\frac{1}{2}} \right) \cr
& - 0.3x = \ln \left( {\frac{1}{2}} \right) \cr
& 0.3x = \ln 2 \cr
& x = \frac{1}{{0.3}}\left( {\ln 2} \right) \cr
& x = \frac{{10}}{3}\left( {\ln 2} \right) \cr} $$
