[tex]3e^{2t\text{ + 5}}\text{ + C}[/tex]
Explanation:[tex]\begin{gathered} \text{Given:} \\ u\text{ = 2t + 5} \\ \int 6e^{2t\text{ + 5}}\text{ dt} \end{gathered}[/tex]
Substituting 2t + 5 with u:
[tex]\begin{gathered} \int 6e^{2t\text{ + 5}}\text{ dt = }\int 6e^u\text{ dt} \\ u\text{ = 2t + 5} \\ \frac{du}{dt}\text{ = 2} \\ du\text{ = 2dt} \\ dt\text{ = du/2} \\ \\ \int 6e^u\text{ dt = }\int 6e^u\text{ }\frac{du}{2} \end{gathered}[/tex][tex]\begin{gathered} \int \frac{6e^udu}{2}\text{ = }\int 3e^udu\text{ } \\ \text{From integration:} \\ \int e^udu\text{ = }e^u \\ 3\int e^udu\text{ = }3(\text{ }e^u)\text{ + C} \\ =3e^{2t\text{ + 5}}\text{ + C} \\ \\ \int 6e^{2t+5}dt\text{ = }3e^{2t\text{ + 5}}\text{ + C} \end{gathered}[/tex]
