Let's evaluate the given integrals step by step.

Problem (a):

$\int_0^1 \frac{2}{1+x^2} \, dx$

Solution:

The integral resembles the derivative of the arctangent function: $\frac{d}{dx} (\arctan(x)) = \frac{1}{1+x^2}.$

So, we can rewrite the integral as: $\int_0^1 \frac{2}{1+x^2} \, dx = 2 \int_0^1 \frac{1}{1+x^2} \, dx.$

The antiderivative of $\frac{1}{1+x^2}$ is $\arctan(x)$, so: $\int_0^1 \frac{2}{1+x^2} \, dx = 2 \left[\arctan(x)\right]_0^1.$

Evaluate the limits: $\arctan(1) = \frac{\pi}{4}, \quad \arctan(0) = 0.$

Thus: $\int_0^1 \frac{2}{1+x^2} \, dx = 2 \left(\frac{\pi}{4} - 0\right) = \frac{\pi}{2}.$

Final Answer for (a):

$\frac{\pi}{2}.$

Problem (b):

$\int_0^1 \sqrt{x(1+3x^2)} \, dx$

Solution:

Let $u = 1+3x^2$, which implies: $du = 6x \, dx \quad \text{or} \quad x \, dx = \frac{du}{6}.$

When $x = 0$, $u = 1$.
When $x = 1$, $u = 4$.

Rewrite the integral: $\int_0^1 \sqrt{x(1+3x^2)} \, dx = \int_1^4 \sqrt{u} \cdot \frac{1}{6} \, du.$

Simplify: $\int_1^4 \sqrt{u} \cdot \frac{1}{6} \, du = \frac{1}{6} \int_1^4 u^{1/2} \, du.$

The antiderivative of $u^{1/2}$ is: $\int u^{1/2} \, du = \frac{2}{3} u^{3/2}.$

So: $\frac{1}{6} \int_1^4 u^{1/2} \, du = \frac{1}{6} \cdot \frac{2}{3} \left[u^{3/2}\right]_1^4.$

Evaluate the limits: $\frac{1}{6} \cdot \frac{2}{3} \left[\left(4^{3/2}\right) - \left(1^{3/2}\right)\right].$

Simplify: $4^{3/2} = \sqrt{4^3} = \sqrt{64} = 8, \quad 1^{3/2} = 1.$

Thus: $\frac{1}{6} \cdot \frac{2}{3} (8 - 1) = \frac{1}{6} \cdot \frac{2}{3} \cdot 7 = \frac{14}{18} = \frac{7}{9}.$

Final Answer for (b):

$\frac{7}{9}.$

Let me know if you'd like further clarification or step-by-step explanations!

Here are 5 related questions to expand your understanding:

1. How does substitution simplify integrals involving products of terms?
2. What is the derivative of $\arctan(x)$, and why does it apply in problem (a)?
3. Why is it important to adjust integration limits during substitution?
4. What are the geometric interpretations of definite integrals like these?
5. How would the answer for problem (b) change if the integral limits were modified?

Tip: Always verify substitution by back-substituting into the original integral to ensure correctness!