Answer
\[\frac{1}{2}\ln \,\left( {{t^2} + 2} \right) + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\frac{t}{{{t^2} + 2}}dt} \hfill \\
Let\,\,u = {t^2} + 2\,\,\,so\,\,that \hfill \\
\,\,\,\,du = 2tdt \hfill \\
\int_{}^{} {\frac{t}{{{t^2} + 2}}dt\,\, = \frac{1}{2}\int_{}^{} {\frac{{2t}}{{{t^2} + 2}}dt} } \hfill \\
\frac{1}{2}\int_{}^{} {\frac{{du}}{u}} \hfill \\
Integrating \hfill \\
\frac{1}{2}\ln \left| u \right| + C \hfill \\
Substituting\,\,u = {t^2} + 2\,\,for\,\,u\,\,gives \hfill \\
\frac{1}{2}\ln \,\left( {{t^2} + 2} \right) + C \hfill \\
\hfill \\
\end{gathered} \]
