We aim to evaluate the integral:

$\int_{0}^{1} \frac{dt}{(t^2+1)(t^2+3)}$

Step 1: Partial Fraction Decomposition

We decompose $\frac{1}{(t^2 + 1)(t^2 + 3)}$ into partial fractions: $\frac{1}{(t^2 + 1)(t^2 + 3)} = \frac{A}{t^2 + 1} + \frac{B}{t^2 + 3}$ Multiplying through by $(t^2 + 1)(t^2 + 3)$, we get: $1 = A(t^2 + 3) + B(t^2 + 1)$ Expanding: $1 = A t^2 + 3A + B t^2 + B$ $1 = (A + B)t^2 + (3A + B)$ Equating coefficients, we have:

1. $A + B = 0$
2. $3A + B = 1$

From $A + B = 0$, we find $B = -A$. Substituting into the second equation: $3A - A = 1 \implies 2A = 1 \implies A = \frac{1}{2}$ $B = -\frac{1}{2}$

Thus, the decomposition is: $\frac{1}{(t^2 + 1)(t^2 + 3)} = \frac{\frac{1}{2}}{t^2 + 1} - \frac{\frac{1}{2}}{t^2 + 3}$

Step 2: Integral Splitting

The integral becomes: $\int_{0}^{1} \frac{dt}{(t^2 + 1)(t^2 + 3)} = \frac{1}{2} \int_{0}^{1} \frac{dt}{t^2 + 1} - \frac{1}{2} \int_{0}^{1} \frac{dt}{t^2 + 3}$

First Integral:

$\int \frac{dt}{t^2 + 1} = \arctan(t)$ Thus: $\int_{0}^{1} \frac{dt}{t^2 + 1} = \arctan(1) - \arctan(0) = \frac{\pi}{4} - 0 = \frac{\pi}{4}$

Second Integral:

$\int \frac{dt}{t^2 + 3} = \frac{1}{\sqrt{3}} \arctan\left(\frac{t}{\sqrt{3}}\right)$ Thus: $\int_{0}^{1} \frac{dt}{t^2 + 3} = \frac{1}{\sqrt{3}} \left[ \arctan\left(\frac{1}{\sqrt{3}}\right) - \arctan(0) \right]$ We know: $\arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}$ So: $\int_{0}^{1} \frac{dt}{t^2 + 3} = \frac{1}{\sqrt{3}} \cdot \frac{\pi}{6} = \frac{\pi}{6\sqrt{3}}$

Step 3: Combine Results

Substitute back into the expression: $\int_{0}^{1} \frac{dt}{(t^2 + 1)(t^2 + 3)} = \frac{1}{2} \cdot \frac{\pi}{4} - \frac{1}{2} \cdot \frac{\pi}{6\sqrt{3}}$ $= \frac{\pi}{8} - \frac{\pi}{12\sqrt{3}}$ Factoring out $\frac{\pi}{8}$: $\int_{0}^{1} \frac{dt}{(t^2 + 1)(t^2 + 3)} = \frac{\pi}{8} \left(1 - \frac{2}{3\sqrt{3}}\right)$

Final Answer:

$\boxed{\frac{\pi}{8}\left(1 - \frac{\sqrt{3}}{6}\right)} \quad \text{(Option 3)}$

Would you like further clarifications or detailed steps? Here are related questions:

1. What are the properties of $\arctan(x)$ used in integration?
2. How do we approach partial fraction decomposition?
3. Can we evaluate similar integrals over infinite limits?
4. What happens if $(t^2 + 3)$ changes to a more general $(t^2 + a^2)$?
5. How can numerical methods verify this result?

Tip: Always verify $\arctan$ values for specific arguments like $\frac{1}{\sqrt{3}}$ or $\sqrt{3}$ to avoid calculation errors.