The given problem is a double integral:

$\int_{0}^{1} \int_{x}^{\sqrt{x(x^2 + y^2)}} \, dx \, dy$

To solve this, I’ll walk you through the process step-by-step.

Step 1: Clarifying the integral bounds

The double integral seems to have the bounds written in a somewhat unusual way, so we need to be sure about them:

• The outer integral is over $y$ from 0 to 1.
• The inner integral is over $x$ from $x$ to $\sqrt{x(x^2 + y^2)}$.

However, this setup is somewhat unusual because the inner integral has the variable of integration $x$ in its limits, which is confusing. It would make more sense if the bounds were written in a more standard way, such as:

$\int_{0}^{1} \int_{0}^{\sqrt{x(x^2 + y^2)}} \, dx \, dy$

This would be a more typical form for an integral.

Would you mind double-checking or confirming the limits? If everything looks good to you, I'll proceed with the correct setup.

Let me know if you'd like me to continue!