Answer
$P = 931265.0973$
Work Step by Step
$$\eqalign{
& c\left( t \right) = 100000 + 4000t,{\text{ }}r = 5\% ,{\text{ }}{t_1} = 10 \cr
& {\text{The present value }}P{\text{ is given by }} \cr
& P = \int_0^{{t_1}} {c\left( t \right)} {e^{ - rt}}dt \cr
& {\text{Substituting}} \cr
& P = \int_0^{10} {\left( {100000 + 4000t} \right)} {e^{ - 0.05t}}dt \cr
& {\text{Integrate by parts}} \cr
& {\text{Let }}u = 100000 + 4000t,{\text{ }}du = 4000dt \cr
& dv = {e^{ - 0.05t}},{\text{ }}v = - 20{e^{ - 0.05t}} \cr
& {\text{By the integration by parts formula}} \cr
& P = \left[ { - 20\left( {100000 + 4000t} \right){e^{ - 0.05t}}} \right]_0^{10} - \int_0^{10} {\left( { - 20{e^{ - 0.05t}}} \right)\left( {4000} \right)dt} \cr
& P = \left[ { - 20\left( {100000 + 4000t} \right){e^{ - 0.05t}}} \right]_0^{10} - 1600000\left[ {{e^{ - 0.05t}}} \right]_0^{10} \cr
& {\text{Evaluating the limits we obtain}} \cr
& P = 301714.1528 + 629550.9445 \cr
& P = 931265.0973 \cr} $$
