$$ \int_0^1\int_0^{x}\!2xy-x\,\mathrm{dydx} $$

To solve the double integral $ \int_0^1 \int_0^x (2xy - x) \, dy \, dx $, we can proceed as follows:

1. **Evaluate the inner integral** with respect to $ y $:
   \[
   \int_0^x (2xy - x) \, dy
   \]

   First, split the integral:
   \[
   \int_0^x 2xy \, dy - \int_0^x x \, dy
   \]

   Now, evaluate each integral separately:

   \[
   \int_0^x 2xy \, dy = 2x \int_0^x y \, dy
   \]
   \[
   2x \left[ \frac{y^2}{2} \right]_0^x = 2x \left( \frac{x^2}{2} - 0 \right) = 2x \cdot \frac{x^2}{2} = x^3
   \]

   Next,
   \[
   \int_0^x x \, dy = x \int_0^x 1 \, dy = x [y]_0^x = x (x - 0) = x^2
   \]

   So, the inner integral becomes:
   \[
   \int_0^x (2xy - x) \, dy = x^3 - x^2
   \]

2. **Evaluate the outer integral** with respect to $ x $:
   \[
   \int_0^1 (x^3 - x^2) \, dx
   \]

   Split the integral again:
   \[
   \int_0^1 x^3 \, dx - \int_0^1 x^2 \, dx
   \]

   Evaluate each integral separately:

   \[
   \int_0^1 x^3 \, dx = \left[ \frac{x^4}{4} \right]_0^1 = \frac{1}{4} - 0 = \frac{1}{4}
   \]

   \[
   \int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1}{3} - 0 = \frac{1}{3}
   \]

   Subtract the results:
   \[
   \int_0^1 (x^3 - x^2) \, dx = \frac{1}{4} - \frac{1}{3} = \frac{3}{12} - \frac{4}{12} = -\frac{1}{12}
   \]

Thus, the value of the double integral is:
\[
\boxed{-\frac{1}{12}}
\]

Do you want details or have any questions?

Here are 5 questions you can ask:
1. How do you perform a double integral with different limits?
2. Can you explain Fubini's theorem and its application in double integrals?
3. How do you change the order of integration in a double integral?
4. What are some practical applications of double integrals in physics?
5. How do you evaluate double integrals over more complex regions?

Tip: Always visualize the region of integration when dealing with double integrals to understand the limits and integrand better.

Learn how to evaluate the double integral ∫∫_(0)^1(2xy - x) dy dx with a detailed step-by-step solution. This problem covers key concepts in double integrals, including limits of integration, the fundamental theorem of calculus, and Fubini's theorem. Suitable for advanced undergraduate students studying calculus.

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