find the indefinite integral.\n int e ^ { 7 - 3 t } d t \n int e ^ { 7 - 3 t } d t =

Explanation:

Step1: Use substitution

Let (u = 7-3t), then (du=-3dt), and (dt=-\frac{1}{3}du).

Step2: Substitute into the integral

[ \begin{align*} \int e^{7 - 3t}dt&=\int e^{u}\left(-\frac{1}{3}\right)du\ &=-\frac{1}{3}\int e^{u}du \end{align*} ]

Step3: Integrate (e^{u})

Since (\int e^{u}du = e^{u}+C), we have (-\frac{1}{3}\int e^{u}du=-\frac{1}{3}e^{u}+C)

Step4: Substitute back (u = 7-3t)

(-\frac{1}{3}e^{u}+C=-\frac{1}{3}e^{7 - 3t}+C)

Answer:

(-\frac{1}{3}e^{7 - 3t}+C)