Answer
$P = 131528.68$
Work Step by Step
$$\eqalign{
& c\left( t \right) = 30000 + 500t,{\text{ }}r = 7\% ,{\text{ }}{t_1} = 5 \cr
& {\text{The present value }}P{\text{ is given by }} \cr
& P = \int_0^{{t_1}} {c\left( t \right)} {e^{ - rt}}dt \cr
& {\text{Substituting}} \cr
& P = \int_0^5 {\left( {30000 + 500t} \right)} {e^{ - 0.07t}}dt \cr
& {\text{Integrate by parts}} \cr
& {\text{Let }}u = 30000 + 500t,{\text{ }}du = 500dt \cr
& dv = {e^{ - 0.07t}},{\text{ }}v = - \frac{{100}}{7}{e^{ - 0.07t}} \cr
& {\text{By the integration by parts formula}} \cr
& P = \left[ { - \frac{{100}}{7}\left( {30000 + 500t} \right){e^{ - 0.07t}}} \right]_0^5 - \int_0^5 {\left( { - \frac{{100}}{7}{e^{ - 0.07t}}} \right)\left( {500} \right)dt} \cr
& P = - \frac{{100}}{7}\left[ {\left( {30000 + 500t} \right){e^{ - 0.07t}}} \right]_0^5 + \frac{{2500000}}{7}\left[ {{e^{ - 0.07t}}} \right]_0^5 \cr
& {\text{Evaluating the limits we obtain}} \cr
& P = 101394.8155 + 105468.53 \cr
& P = 131528.68 \cr} $$
