Answer
$${\text{where }}{\bf{C}}{\text{ is a vector constant of integration}}$$
Work Step by Step
$$\eqalign{
& \int {\left\langle {t{e^t},\ln t} \right\rangle } dt \cr
& {\text{Integrating}} \cr
& \int {\left\langle {t{e^t},\ln t} \right\rangle } dt = \left\langle {\int {t{e^t}dt} ,\int {\ln tdt} } \right\rangle \cr
& {\text{Integrating by parts we obtain}} \cr
& \left\langle {t{e^t} - {e^t},t\ln t - 1} \right\rangle + {\bf{C}} \cr
& {\text{where }}{\bf{C}}{\text{ is a vector constant of integration}} \cr} $$
