Question

Perform the substitution to evaluate the indefinite integral. use C for the constant of integration

Answer 1

`3e^{2t\text{ + 5}}\text{ + C}`

Step-by-step explanation:
`\begin{gathered} \text{Given:} \\ u\text{ = 2t + 5} \\ \int 6e^{2t\text{ + 5}}\text{ dt} \end{gathered}`

Substituting 2t + 5 with u:

`\begin{gathered} \int 6e^{2t\text{ + 5}}\text{ dt = }\int 6e^u\text{ dt} \\ u\text{ = 2t + 5} \\ (du)/(dt)\text{ = 2} \\ du\text{ = 2dt} \\ dt\text{ = du/2} \\ \\ \int 6e^u\text{ dt = }\int 6e^u\text{ }(du)/(2) \end{gathered}`
`\begin{gathered} \int (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}`