Interactive Differential Equations: A Step-by-Step Approach to Methods & Modeling

Subsection B.2.2 Example: Applying Integration by Parts

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:

$$\int t{e}^{t}dt=t{e}^t-\int e^{t}dt\text{.}$$

Step 3: Solve the remaining integral:

$$\int e^{t}dt=e^t\text{.}$$

Step 4: Combine the results:

$$t{e}^t-e^t+C\text{.}$$