To evaluate the integral

$I = \int_{0}^{3} x e^{x^2} \, dx,$

we recognize that the term $x e^{x^2}$ suggests a substitution approach. Here's the step-by-step solution:

Step 1: Substitution

Let: $u = x^2, \quad \text{so that} \quad du = 2x \, dx \quad \text{or} \quad x \, dx = \frac{1}{2} \, du.$

Change the limits accordingly:

• When $x = 0$, $u = 0^2 = 0$.
• When $x = 3$, $u = 3^2 = 9$.

The integral becomes: $I = \int_{0}^{9} e^u \cdot \frac{1}{2} \, du = \frac{1}{2} \int_{0}^{9} e^u \, du.$

Step 2: Evaluate the Integral

The integral of $e^u$ is: $\int e^u \, du = e^u.$

Thus: $I = \frac{1}{2} \left[ e^u \right]_{0}^{9}.$

Step 3: Apply the Limits

Substitute the limits into $e^u$: $I = \frac{1}{2} \left( e^9 - e^0 \right).$

Since $e^0 = 1$, this simplifies to: $I = \frac{1}{2} \left( e^9 - 1 \right).$

Final Answer:

$\boxed{\frac{1}{2} \left( e^9 - 1 \right)}$

Would you like additional details or help with related problems?

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Related Questions:

1. What is the integral of $x e^{x^2}$ over an arbitrary interval $[a, b]$?
2. How do you numerically approximate this integral if $e^9$ is too large for exact calculation?
3. What happens if the integral is $\int x e^{-x^2} dx$?
4. How do substitution techniques compare to integration by parts in such problems?
5. Can this integral be evaluated using series expansion?

Tip:

When handling integrals with terms like $x \cdot f(x^2)$, substitution with $u = x^2$ is often the simplest and most effective method.