To evaluate the integral $R = \int_{0}^{2} x e^x \, dx$, we can use the technique of integration by parts.

Step 1: Identify $u$ and $dv$

For integration by parts, we use the formula: $\int u \, dv = uv - \int v \, du$ Let $u = x$, so that $du = dx$.
Let $dv = e^x \, dx$, so that $v = e^x$.

Step 2: Apply the Integration by Parts Formula

Now, apply the formula: $\int x e^x \, dx = x e^x - \int e^x \, dx$ The integral of $e^x$ with respect to $x$ is simply $e^x$, so we have: $\int x e^x \, dx = x e^x - e^x + C$ where $C$ is the constant of integration.

Step 3: Evaluate the Definite Integral

Now, we need to evaluate this expression from $0$ to $2$: $R = \left[ x e^x - e^x \right]_{0}^{2}$ Substituting the limits: $R = \left( 2e^2 - e^2 \right) - \left( 0 \cdot e^0 - e^0 \right)$ Simplify this expression: $R = e^2 - (-1) = e^2 + 1$

So, the evaluated integral is: $R = e^2 + 1$

Would you like more details or have any questions?

Related Questions:

1. How does integration by parts work with trigonometric functions?
2. What if the integrand was $x^2 e^x$? How would that change the approach?
3. How does integration by parts compare to substitution?
4. What is the geometric interpretation of definite integrals?
5. How would this integral change if the integrand was $xe^{-x}$?

Tip: When using integration by parts, choosing $u$ as a function that simplifies upon differentiation and $dv$ as a function that is easy to integrate can simplify the process.