The integral in the image is a double integral:

$\int_0^1 \int_{x}^{3} e^{x^2} \, dy \, dx$

Step-by-Step Solution:

Step 1: Evaluate the inner integral

The inner integral is with respect to $y$, treating $x$ as a constant: $\int_x^3 e^{x^2} \, dy$

Since $e^{x^2}$ does not depend on $y$, it is treated as a constant during the integration: $\int_x^3 e^{x^2} \, dy = e^{x^2} \int_x^3 1 \, dy = e^{x^2} \cdot \left[ y \right]_x^3$

Substituting the limits of $y$: $e^{x^2} \cdot \left[ y \right]_x^3 = e^{x^2} \cdot \left(3 - x\right)$

So the result of the inner integral is: $e^{x^2} \cdot (3 - x)$

Step 2: Evaluate the outer integral

The outer integral is: $\int_0^1 e^{x^2} \cdot (3 - x) \, dx$

Distribute $e^{x^2}$: $\int_0^1 \left[ 3e^{x^2} - x e^{x^2} \right] \, dx$

This can be separated into two integrals: $\int_0^1 3e^{x^2} \, dx - \int_0^1 x e^{x^2} \, dx$

Step 3: Solve each part

1. For $\int_0^1 3e^{x^2} \, dx$:
   This integral does not have a closed-form solution and must be left as is or evaluated numerically.

   $\int_0^1 3e^{x^2} \, dx \quad (\text{numerical evaluation needed if exact value is required})$

2. For $\int_0^1 x e^{x^2} \, dx$:
   Let $u = x^2$, so $du = 2x \, dx$ or $\frac{1}{2} du = x \, dx$.
   When $x = 0$, $u = 0$.
   When $x = 1$, $u = 1$.
   Substituting into the integral: $\int_0^1 x e^{x^2} \, dx = \frac{1}{2} \int_0^1 e^u \, du$

   The integral of $e^u$ is simply $e^u$: $\frac{1}{2} \int_0^1 e^u \, du = \frac{1}{2} \left[ e^u \right]_0^1 = \frac{1}{2} \left( e^1 - e^0 \right) = \frac{1}{2} (e - 1)$

Step 4: Combine the results

The final result is: $\int_0^1 \int_x^3 e^{x^2} \, dy \, dx = 3 \int_0^1 e^{x^2} \, dx - \frac{1}{2} (e - 1)$

Final Answer

$\boxed{\int_0^1 \int_x^3 e^{x^2} \, dy \, dx = 3 \int_0^1 e^{x^2} \, dx - \frac{1}{2} (e - 1)}$

Would you like a numerical approximation for the result or further explanation on any step?

Follow-up Questions:

1. What is the numerical approximation for $\int_0^1 \int_x^3 e^{x^2} \, dy \, dx$?
2. How is substitution used in integrals involving exponential functions?
3. Why does $\int_0^1 e^{x^2} \, dx$ lack a closed-form solution?
4. What are some techniques for solving double integrals?
5. How would the bounds change the approach if reversed?

Tip:

For integrals involving $e^{x^2}$, numerical methods like Simpson's rule or Gaussian quadrature are often necessary for practical evaluations.