Find the definite integral of int 0 2 e frac x ... - SS3 Mathematics Integral Calculus (Integration) Question

First integrate \(\int_{}^{}e^{\frac{x}{2}}dx\), let \(u = \frac{x}{2}\), \(\frac{du}{dx} = \frac{2(1) - x(0)}{4} = \frac{2}{4} = \frac{1}{2}\), \(dx = 4\ du\)

\[\int_{}^{}e^{u}4\ du = 4\int_{}^{}e^{u}du = 4\ e^{u} = 4e^{\frac{x}{2}}\ \]

\[\int_{0}^{2}e^{\frac{x}{2}}dx = \ \left\lbrack 4e^{\frac{x}{2}} \right\rbrack_{0}^{2} = \left\lbrack 4e^{\frac{2}{2}} \right\rbrack - \left\lbrack 4e^{\frac{0}{2}} \right\rbrack = 4e - 4(1) = 4e - 4 = 4(e - 1)\]