We are tasked with evaluating the definite integral

$I(x) = \int_1^{\sqrt{x}} \frac{t^2}{2 + t^4} \, dt$

Step 1: Substitution and Simplification

We can simplify the integral by using a substitution to handle the dependence on $x$. Since the upper limit of the integral involves $\sqrt{x}$, a substitution based on $u = t^2$ is likely useful. Let’s proceed with that:

1. Let $u = t^2$. Then, $du = 2t \, dt$, or equivalently, $t \, dt = \frac{du}{2}$.
2. The limits change as follows:
   • When $t = 1$, $u = 1^2 = 1$,
   • When $t = \sqrt{x}$, $u = (\sqrt{x})^2 = x$.

Thus, the integral becomes:

$I(x) = \int_1^x \frac{(u^{1/2})^2}{2 + u^2} \cdot \frac{du}{2}$

Simplifying the integrand:

$I(x) = \int_1^x \frac{u}{2 + u^2} \cdot \frac{du}{2}$

$I(x) = \frac{1}{2} \int_1^x \frac{u}{2 + u^2} \, du$

Step 2: Integration of the New Integral

To integrate $\frac{u}{2 + u^2}$, we can use a standard substitution:

1. Let $v = 2 + u^2$, so $dv = 2u \, du$, or equivalently, $u \, du = \frac{dv}{2}$.
2. The limits change as follows:
   • When $u = 1$, $v = 2 + 1^2 = 3$,
   • When $u = x$, $v = 2 + x^2$.

Substituting into the integral:

$I(x) = \frac{1}{2} \int_3^{2 + x^2} \frac{1}{v} \cdot \frac{dv}{2}$

Simplifying:

$I(x) = \frac{1}{4} \int_3^{2 + x^2} \frac{dv}{v}$

Step 3: Final Integration

The integral $\int \frac{1}{v} \, dv$ is straightforward, and we get:

$I(x) = \frac{1}{4} \left[ \ln|v| \right]_3^{2 + x^2}$

Substituting the limits:

$I(x) = \frac{1}{4} \left( \ln(2 + x^2) - \ln(3) \right)$

Final Answer

Thus, the integral evaluates to:

$I(x) = \frac{1}{4} \ln \left( \frac{2 + x^2}{3} \right)$

Do you have any questions or would you like more details on any step? Here are some related questions:

1. What are some common substitutions used in definite integrals?
2. How do you evaluate integrals with variable upper limits?
3. What are some techniques for solving integrals with rational functions?
4. How can we handle logarithmic expressions in integrals?
5. How do you check the correctness of your integration steps?

Tip: For integrals with variable limits, carefully consider substitutions that simplify both the integrand and the limits of integration.