We aim to evaluate the double integral:

$\int_0^1 \int_0^{\sqrt{1+x^2}} \frac{1}{1+x^2 + y^2} \, dy \, dx.$

Steps to Solve:

Step 1: Simplify the inner integral

The inner integral is: $\int_0^{\sqrt{1+x^2}} \frac{1}{1+x^2 + y^2} \, dy.$

Let: $u = 1 + x^2 + y^2 \quad \text{so that} \quad du = 2y \, dy.$

When $y = 0$, $u = 1 + x^2$.

When $y = \sqrt{1+x^2}$, $u = 2 + x^2$.

Substitute and adjust for the factor of 2: $\int_0^{\sqrt{1+x^2}} \frac{1}{1+x^2 + y^2} \, dy = \frac{1}{2} \int_{1+x^2}^{2+x^2} \frac{1}{u} \, du.$

Step 2: Evaluate the $u$-integral

The integral of $\frac{1}{u}$ is $\ln |u|$. Thus: $\int_0^{\sqrt{1+x^2}} \frac{1}{1+x^2 + y^2} \, dy = \frac{1}{2} \left[ \ln(2+x^2) - \ln(1+x^2) \right].$

Simplify the logarithmic difference: $\frac{1}{2} \left[ \ln(2+x^2) - \ln(1+x^2) \right] = \frac{1}{2} \ln \left(\frac{2+x^2}{1+x^2}\right).$

Step 3: Outer integral

Now substitute back into the outer integral: $\int_0^1 \int_0^{\sqrt{1+x^2}} \frac{1}{1+x^2 + y^2} \, dy \, dx = \int_0^1 \frac{1}{2} \ln \left(\frac{2+x^2}{1+x^2}\right) \, dx.$

Factor out $\frac{1}{2}$: $\frac{1}{2} \int_0^1 \ln \left(\frac{2+x^2}{1+x^2}\right) \, dx.$

Step 4: Simplify the argument of the logarithm

Let: $\ln \left(\frac{2+x^2}{1+x^2}\right) = \ln(2+x^2) - \ln(1+x^2).$

So: $\int_0^1 \ln \left(\frac{2+x^2}{1+x^2}\right) \, dx = \int_0^1 \ln(2+x^2) \, dx - \int_0^1 \ln(1+x^2) \, dx.$

Step 5: Evaluate each term

1. First term: $\int_0^1 \ln(2+x^2) \, dx$ Use substitution $x^2 = t$, $2x \, dx = dt$. Bounds change: when $x = 0, t = 0$; when $x = 1, t = 1$. $\int_0^1 \ln(2+x^2) \, dx = \frac{1}{2} \int_0^1 \ln(2+t) \, \frac{dt}{\sqrt{t}}.$

2. Second term: $\int_0^1 \ln(1+x^2) \, dx$ Use the same substitution $x^2 = t$, $2x \, dx = dt$, bounds $t \in [0, 1]$: $\int_0^1 \ln(1+x^2) \, dx = \frac{1}{2} \int_0^1 \ln(1+t) \, \frac{dt}{\sqrt{t}}.$

These integrals require special functions or numerical evaluation. If you'd like, I can proceed with exact computations or approximations.

Let me know how detailed you'd like the final evaluation!

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Relative Questions:

1. How do substitution techniques simplify integral calculations?
2. What are common methods to handle nested integrals in multivariable calculus?
3. How does logarithmic integration relate to exponential functions?
4. How would changing the bounds affect this specific integral?
5. What is the role of numerical integration when handling complex expressions?

Tip: Always check the feasibility of substitution or symmetry in integrals before jumping into complex methods.