Subsection B.2.2 Example: Applying Integration by Parts
Consider the integral:
\begin{equation*}
\int t \, e^t \, dt \text{.}
\end{equation*}
We’ll apply integration by parts, following these steps:
- Step 1: Identify \(u\) and \(dv\text{.}\) In this case, we choose \(u = t\) and \(dv = e^t \, dt\text{.}\) This makes \(du = dt\) and \(v = e^t\text{.}\)
- Step 2: Substitute into the integration by parts formula:
- \begin{equation*} \int t \, e^t \, dt = t \, e^t - \int e^t \, dt \text{.} \end{equation*}
- Step 3: Solve the remaining integral:
- \begin{equation*} \int e^t \, dt = e^t \text{.} \end{equation*}
- Step 4: Combine the results:
- \begin{equation*} t \, e^t - e^t + C \text{.} \end{equation*}
And that’s the final result:
\begin{equation*}
\int t \, e^t \, dt = t \, e^t - e^t + C \text{.}
\end{equation*}
