calculate the double integral of x*e^(xy) over R where R = [0,1]x[1,2]

To calculate the double integral of \( x \cdot e^{xy} \) over the region \( R = [0,1] \times [1,2] \), we can write the integral as:

\[
\int_1^2 \int_0^1 x \cdot e^{xy} \, dx \, dy
\]

### Step-by-Step Calculation:

1. **Evaluate the inner integral:**

   We first integrate with respect to \( x \). The inner integral is:
   \[
   \int_0^1 x \cdot e^{xy} \, dx
   \]
   To solve this, we use integration by parts. Let:
   - \( u = x \), so \( du = dx \),
   - \( dv = e^{xy} \, dx \), so \( v = \frac{e^{xy}}{y} \).

   Now, apply the integration by parts formula:
   \[
   \int u \, dv = uv - \int v \, du
   \]
   Therefore:
   \[
   \int_0^1 x \cdot e^{xy} \, dx = \left[ \frac{x \cdot e^{xy}}{y} \right]_0^1 - \int_0^1 \frac{e^{xy}}{y} \, dx
   \]
   
   Evaluating the first term:
   \[
   \left[ \frac{x \cdot e^{xy}}{y} \right]_0^1 = \frac{e^y}{y} - 0 = \frac{e^y}{y}
   \]

   Now for the second term:
   \[
   \int_0^1 \frac{e^{xy}}{y} \, dx = \frac{1}{y} \int_0^1 e^{xy} \, dx
   \]
   Since \( \int e^{xy} \, dx = \frac{e^{xy}}{y} \), we have:
   \[
   \frac{1}{y} \left[ \frac{e^{xy}}{y} \right]_0^1 = \frac{1}{y} \left( \frac{e^y}{y} - \frac{1}{y} \right)
   \]
   Simplifying this expression:
   \[
   \frac{e^y - 1}{y^2}
   \]

   Putting everything together, the inner integral becomes:
   \[
   \frac{e^y}{y} - \frac{e^y - 1}{y^2} = \frac{e^y (y - 1) + 1}{y^2}
   \]

2. **Evaluate the outer integral:**

   Now, we need to evaluate the outer integral:
   \[
   \int_1^2 \frac{e^y (y - 1) + 1}{y^2} \, dy
   \]
   This integral can be split into two parts:
   \[
   \int_1^2 \frac{e^y (y - 1)}{y^2} \, dy + \int_1^2 \frac{1}{y^2} \, dy
   \]

   Let's calculate each part:

   - The second part \( \int_1^2 \frac{1}{y^2} \, dy \) is straightforward:
     \[
     \int_1^2 \frac{1}{y^2} \, dy = \left[ -\frac{1}{y} \right]_1^2 = -\frac{1}{2} + 1 = \frac{1}{2}
     \]

   - The first part \( \int_1^2 \frac{e^y (y - 1)}{y^2} \, dy \) requires substitution or numerical integration. We can approximate this term.

3. **Final Result:**

   Combining both parts, the final result for the integral is approximately:

   \[
   \int_1^2 \frac{e^y (y - 1)}{y^2} \, dy + \frac{1}{2}
   \]

This integral can be further simplified or calculated numerically for an exact value.

Would you like more details or a numerical approximation?

Here are 5 related questions to expand on this concept:
1. How is integration by parts applied to solve integrals?
2. How do you evaluate improper integrals using limits?
3. What methods can be used to solve double integrals with complex regions?
4. How does changing the order of integration affect the result of a double integral?
5. How do numerical methods help in evaluating difficult integrals?

**Tip:** When using integration by parts, always carefully choose \( u \) and \( dv \) to simplify the problem in subsequent steps.

Explore how to calculate the double integral of x*e^(xy) over the region [0,1] x [1,2] using integration by parts. This detailed solution includes step-by-step calculations, handling exponential functions, and applying the integration by parts technique. Suitable for undergraduate calculus students.

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