Define algebraically \((\overline{I a})_{n}\)

Question

i. Define algebraically \((\overline{I a})_{n}\)

ii. Hence, prove that \((\overline{I a})_{n}=\frac{\bar{a}_{\eta}-n v^{n}}{\delta}\).

Answer 1

(i)
\((\overline{I a})_{n}=\int_{0}^{n} t v^{t} d t\)
OR
\[
(\overline{I a})_{\pi}=\int_{0}^{n} t e^{-\delta t} d t
\]
(ii)
\[
(\overline{I a})_{n}=\int_{0}^{n} t v^{t} d t=\int_{0}^{n} t e^{-\delta t} d t
\]
Integrating by parts:
\[
\begin{aligned}
&u=t \\
&\frac{d v}{d t}=e^{-\delta t}
\end{aligned}
\]
\[
(\overline{I a})_{\pi}=\left[t\left(\frac{1}{-\delta} e^{-\delta t}\right)\right]_{0}^{n}-\int_{0}^{n} \frac{1}{-\delta} e^{-\delta t} d t=-\frac{n v^{n}}{\delta}+\frac{1}{\delta} \int_{0}^{n} v^{t} d t=-\frac{n v^{n}}{\delta}+\frac{\bar{a}_{\eta}}{\delta}
\]
\[
=\frac{\bar{a}_{\eta}-n v^{n}}{\delta}
\]