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

Step 1: Substitution

Let $u = x^2$. Then, the derivative of $u$ with respect to $x$ is: $\frac{du}{dx} = 2x$ So, $du = 2x \, dx$, or $\frac{1}{2} du = x \, dx$.

Step 2: Rewrite the Integral

Substitute into the integral: $R = \int_{0}^{2} x e^{x^2} \, dx = \int_{u(0)}^{u(2)} e^{u} \cdot \frac{1}{2} du$ Given $u = x^2$, when $x = 0$, $u = 0$, and when $x = 2$, $u = 4$. Therefore, the integral becomes: $R = \frac{1}{2} \int_{0}^{4} e^u \, du$

Step 3: Evaluate the Integral

The integral of $e^u$ with respect to $u$ is simply $e^u$. So, we have: $R = \frac{1}{2} \left[ e^u \right]_{0}^{4} = \frac{1}{2} \left( e^4 - e^0 \right)$

Step 4: Simplify the Result

Since $e^0 = 1$, the final expression is: $R = \frac{1}{2} \left( e^4 - 1 \right)$

So, the evaluated integral is: $R = \frac{e^4 - 1}{2}$

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Related Questions:

1. How would the integral change if the limits were different, say from 1 to 3?
2. Can the method of integration by parts be applied to this integral?
3. How does substitution work in integrals involving trigonometric functions?
4. What is the significance of the constant factor in definite integrals?
5. How does the exponential function behave under integration compared to polynomial functions?

Tip: Substitution is a powerful technique for solving integrals where the integrand is a composite function, especially when one part of the function is the derivative of another.