$$4\int_{0}^{1} \frac{1}{\sqrt{3 + x^2} + \sqrt{1 + x^2}} \, dx - 3\log_e(\sqrt{3})$$ is equal to :

To find the value of the given expression, we first rewrite the integrand by rationalizing the denominator.

The given integral is:

$$I = \int_{0}^{1} \frac{1}{\sqrt{3 + x^2} + \sqrt{1 + x^2}} \, dx$$

Multiplying the numerator and the denominator by the conjugate expression $$\sqrt{3 + x^2} - \sqrt{1 + x^2}}$$:

$$\frac{1}{\sqrt{3 + x^2} + \sqrt{1 + x^2}} = \frac{\sqrt{3 + x^2} - \sqrt{1 + x^2}}{(3 + x^2) - (1 + x^2)} = \frac{\sqrt{3 + x^2} - \sqrt{1 + x^2}}{2}$$

Substituting this back into the expression simplifies it to:

$$4 \times \frac{1}{2} \int_{0}^{1} \left(\sqrt{3 + x^2} - \sqrt{1 + x^2}\right) \, dx - 3\log_e(\sqrt{3})$$

$$= 2\int_{0}^{1} \sqrt{3 + x^2} \, dx - 2\int_{0}^{1} \sqrt{1 + x^2} \, dx - 3\log_e(\sqrt{3})$$

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Step 1: Evaluate the first integral using the formula $$\int \sqrt{a^2 + x^2} \, dx = \frac{x}{2}\sqrt{a^2 + x^2} + \frac{a^2}{2}\log_e\left|x + \sqrt{a^2 + x^2}\right|$$

For the first part where $$a^2 = 3$$:

$$\int_{0}^{1} \sqrt{3 + x^2} \, dx = \left[ \frac{x}{2}\sqrt{3 + x^2} + \frac{3}{2}\log_e\left|x + \sqrt{3 + x^2}\right| \right]_{0}^{1}$$

$$= \left(\frac{1}{2}\sqrt{4} + \frac{3}{2}\log_e(1 + 2)\right) - \left(0 + \frac{3}{2}\log_e(\sqrt{3})\right)$$

$$= 1 + \frac{3}{2}\log_e(3) - \frac{3}{2}\log_e(\sqrt{3})$$

Since $$\log_e(3) = 2\log_e(\sqrt{3})$$, this simplifies to:

$$= 1 + 3\log_e(\sqrt{3}) - \frac{3}{2}\log_e(\sqrt{3}) = 1 + \frac{3}{2}\log_e(\sqrt{3})$$

Multiplying by 2 gives:

$$2\int_{0}^{1} \sqrt{3 + x^2} \, dx = 2 + 3\log_e(\sqrt{3})$$

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Step 2: Evaluate the second integral where $$a^2 = 1$$

$$\int_{0}^{1} \sqrt{1 + x^2} \, dx = \left[ \frac{x}{2}\sqrt{1 + x^2} + \frac{1}{2}\log_e\left|x + \sqrt{1 + x^2}\right| \right]_{0}^{1}$$

$$= \left(\frac{1}{2}\sqrt{2} + \frac{1}{2}\log_e(1 + \sqrt{2})\right) - 0 = \frac{\sqrt{2}}{2} + \frac{1}{2}\log_e(1 + \sqrt{2})$$

Multiplying by 2 gives:

$$2\int_{0}^{1} \sqrt{1 + x^2} \, dx = \sqrt{2} + \log_e(1 + \sqrt{2})$$

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Step 3: Combine all evaluations into the main expression

Substituting the values from Step 1 and Step 2:

$$Required value = \left(2 + 3\log_e(\sqrt{3})\right) - \left(\sqrt{2} + \log_e(1 + \sqrt{2})\right) - 3\log_e(\sqrt{3})$$

$$Required value = 2 - \sqrt{2} - \log_e(1 + \sqrt{2})$$