https://file.aiquickdraw.com/mathbot/file/1732174440603rpm5fgxx.jpg

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.

Evaluate the double integral \( \int_0^1 \int_x^3 e^{x^2} \, dy \, dx \).

Step-by-step solution for the double integral \( \int_0^1 \int_x^3 e^{x^2} \, dy \, dx \). This problem covers double integration, exponential functions, and substitution techniques, suitable for advanced calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation