To evaluate the definite integral $\int_0^2 t e^{-t} \, dt$ using integration by parts, we follow these steps:

Step 1: Recall the formula for integration by parts

$\int u \, dv = uv - \int v \, du$ We need to choose $u$ and $dv$ from the given integrand $t e^{-t}$.

Step 2: Assign $u$ and $dv$

Let:

• $u = t$ (so that $du = dt$)
• $dv = e^{-t} dt$ (so that $v = -e^{-t}$)

Step 3: Apply the integration by parts formula

$\int t e^{-t} \, dt = u v - \int v \, du$ Substitute the expressions for $u$, $v$, $du$, and $dv$: $\int t e^{-t} \, dt = t (-e^{-t}) - \int (-e^{-t}) \, dt$ $= -t e^{-t} + \int e^{-t} \, dt$ Now, integrate $\int e^{-t} \, dt$: $\int e^{-t} \, dt = -e^{-t}$ Thus, the expression becomes: $-t e^{-t} - e^{-t} + C$ where $C$ is the constant of integration.

Step 4: Evaluate the definite integral from 0 to 2

Now, we evaluate the expression at the limits 0 and 2: $\int_0^2 t e^{-t} \, dt = \left[ -t e^{-t} - e^{-t} \right]_0^2$

At the upper limit ($t = 2$): $-2 e^{-2} - e^{-2} = -3e^{-2}$

At the lower limit ($t = 0$): $-0 e^{0} - e^{0} = -1$

Step 5: Final result

Now subtract the result at the lower limit from the result at the upper limit: $-3e^{-2} - (-1) = -3e^{-2} + 1$ Thus, the value of the definite integral is: $\int_0^2 t e^{-t} \, dt = 1 - 3e^{-2}$

Would you like any further details or clarification on this process?

Here are 5 related questions:

1. How do you choose $u$ and $dv$ in integration by parts?
2. How can you simplify the result further if given an approximate value of $e^{-2}$?
3. What are some cases where integration by parts doesn't simplify the integral?
4. How do you apply integration by parts to definite integrals in other forms?
5. Can we use other methods like substitution for this integral?

Tip: Integration by parts works well when one part of the integrand simplifies upon differentiation (like $t$ here).