What is $\frac{1}{2} \int {e^{\frac{t}{2}}dt}$ equal to? Would using substitution be helpful to get rid of the exponent of the variable?
I tried substituting "$u$" in but it did not seem to help finding the integral.
Yes, it would be helpful.

$$u = t/2$$
$$du = \frac{1}{2}dt \Longrightarrow dt = 2du$$

And so:

$$\int e^{t/2} dt = 2\int e^{u}du$$

Of course, the exponential function is its own derivative, and it's own integral up to an integration constant.
$\frac 1 2 \int e^\frac t 2 dt$

= 1/2[e^(t/2)/(1/2)+C] 

= e^(t/2)+C/2 

= e^(t/2) + C [C-Integral Constant]