We are tasked with evaluating the double integral:

$\int_0^1 \int_{\sqrt{x}}^1 e^{\frac{x}{y}} \, dy \, dx$

Step-by-step solution:

1. Inner Integral (with respect to $y$): $I_1 = \int_{\sqrt{x}}^1 e^{\frac{x}{y}} \, dy$

   To solve this, let's use the substitution method:

   • Let $u = \frac{x}{y}$, which gives $du = -\frac{x}{y^2} dy$, and rearranging gives $dy = -\frac{y^2}{x} du$.

   Substituting the bounds:

   • When $y = \sqrt{x}$, $u = \frac{x}{\sqrt{x}} = \sqrt{x}$.
   • When $y = 1$, $u = x$.

2. Outer Integral (with respect to $x$):

   After finding the result of the inner integral, we can integrate with respect to $x$ over the interval $[0, 1]$.

Let me proceed to calculate the double integral.The value of the double integral

$\int_0^1 \int_{\sqrt{x}}^1 e^{\frac{x}{y}} \, dy \, dx$

is $\frac{1}{2}$.

Would you like further details or have any questions?

Here are some related problems to explore:

1. Evaluate $\int_0^1 \int_0^x e^{x+y} \, dy \, dx$.
2. Find the double integral $\int_0^1 \int_0^x y^2 \, dy \, dx$.
3. Compute $\int_0^1 \int_0^1 xy \, dy \, dx$.
4. Solve $\int_0^2 \int_0^x \sin(y) \, dy \, dx$.
5. Determine $\int_0^1 \int_0^1 e^{x+y} \, dy \, dx$.

Tip: When dealing with nested integrals, always integrate step-by-step, starting from the inner integral.