Question

$$ \int \frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}} \, dx - 3 \log \left( \sqrt{3} \right) $$ is equal to:

• A. \( 2 + \sqrt{2} + \log \left( 1 + \sqrt{2} \right) \)
• B. \( 2 - \sqrt{2} - \log \left( 1 + \sqrt{2} \right) \)
• C. \( 2 + \sqrt{2} - \log \left( 1 + \sqrt{2} \right) \)
• D. \( 2 - \sqrt{2} + \log \left( 1 + \sqrt{2} \right) \)

Hint:

When solving integrals with square roots, rationalizing the denominator by multiplying numerator and denominator by the conjugate expression can often simplify the problem significantly. This technique is particularly useful for integrals involving sums of square roots.

Answer 1

The Correct Option is B

The objective is to determine the value of the expression \( \int \frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}} \, dx - 3 \log \left( \sqrt{3} \right) \). The presence of constant options suggests a definite integral. Analysis of these options indicates that the problem likely involved an integral from -1 to 1 with a numerator factor of 2. This inferred problem will be solved.

The expression to evaluate is:

\[ E = \int_{-1}^{1} \frac{2}{\sqrt{3+x^2}+\sqrt{1+x^2}} \, dx - 3 \log \left( \sqrt{3} \right) \]

Core Concept:

Rationalization of the denominator is used to simplify the integral. The resulting integral is then evaluated using the standard integral formula for \( \sqrt{a^2+x^2} \):

\[ \int \sqrt{a^2+x^2} \, dx = \frac{x}{2}\sqrt{a^2+x^2} + \frac{a^2}{2} \log \left| x + \sqrt{a^2+x^2} \right| + C \]

Solution Steps:

Step 1: Evaluate the integral component, \( I = \int_{-1}^{1} \frac{2}{\sqrt{3+x^2}+\sqrt{1+x^2}} \, dx \), by rationalizing the integrand.

\[ \frac{2}{\sqrt{3+x^2}+\sqrt{1+x^2}} \times \frac{\sqrt{3+x^2}-\sqrt{1+x^2}}{\sqrt{3+x^2}-\sqrt{1+x^2}} = \frac{2(\sqrt{3+x^2}-\sqrt{1+x^2})}{(3+x^2)-(1+x^2)} \] \[ = \frac{2(\sqrt{3+x^2}-\sqrt{1+x^2})}{2} = \sqrt{3+x^2}-\sqrt{1+x^2} \]

Step 2: The integral is simplified to:

\[ I = \int_{-1}^{1} (\sqrt{3+x^2} - \sqrt{1+x^2}) \, dx \]

Step 3: Determine the antiderivatives using the standard formula.

For \( \int \sqrt{3+x^2} dx \) (\( a^2 = 3 \)):

\[ \frac{x}{2}\sqrt{3+x^2} + \frac{3}{2} \log(x + \sqrt{3+x^2}) \]

For \( \int \sqrt{1+x^2} dx \) (\( a^2 = 1 \)):

\[ \frac{x}{2}\sqrt{1+x^2} + \frac{1}{2} \log(x + \sqrt{1+x^2}) \]

Step 4: Combine the antiderivatives to find \(F(x)\) for \( \sqrt{3+x^2} - \sqrt{1+x^2} \).

\[ F(x) = \left( \frac{x}{2}\sqrt{3+x^2} + \frac{3}{2} \log(x + \sqrt{3+x^2}) \right) - \left( \frac{x}{2}\sqrt{1+x^2} + \frac{1}{2} \log(x + \sqrt{1+x^2}) \right) \]

Step 5: Evaluate \( F(x) \) at the integration limits \(x=1\) and \(x=-1\).

At \( x = 1 \):

\[ F(1) = \left( \frac{1}{2}\sqrt{3+1} + \frac{3}{2} \log(1 + \sqrt{3+1}) \right) - \left( \frac{1}{2}\sqrt{1+1} + \frac{1}{2} \log(1 + \sqrt{1+1}) \right) \] \[ F(1) = \left( 1 + \frac{3}{2} \log(3) \right) - \left( \frac{\sqrt{2}}{2} + \frac{1}{2} \log(1 + \sqrt{2}) \right) = 1 + \frac{3}{2}\log(3) - \frac{\sqrt{2}}{2} - \frac{1}{2}\log(1+\sqrt{2}) \]

At \( x = -1 \):

\[ F(-1) = \left( \frac{-1}{2}\sqrt{3+1} + \frac{3}{2} \log(-1 + \sqrt{3+1}) \right) - \left( \frac{-1}{2}\sqrt{1+1} + \frac{1}{2} \log(-1 + \sqrt{1+1}) \right) \] \[ F(-1) = \left( -1 + \frac{3}{2} \log(1) \right) - \left( -\frac{\sqrt{2}}{2} + \frac{1}{2} \log(\sqrt{2}-1) \right) \]

Using \( \log(1) = 0 \) and \( \log(\sqrt{2}-1) = -\log(\sqrt{2}+1) \):

\[ F(-1) = -1 - \left( -\frac{\sqrt{2}}{2} - \frac{1}{2} \log(1+\sqrt{2}) \right) = -1 + \frac{\sqrt{2}}{2} + \frac{1}{2}\log(1+\sqrt{2}) \]

Step 6: Calculate the definite integral \( I = F(1) - F(-1) \).

\[ I = \left( 1 + \frac{3}{2}\log(3) - \frac{\sqrt{2}}{2} - \frac{1}{2}\log(1+\sqrt{2}) \right) - \left( -1 + \frac{\sqrt{2}}{2} + \frac{1}{2}\log(1+\sqrt{2}) \right) \] \[ I = 2 - \sqrt{2} + \frac{3}{2}\log(3) - \log(1+\sqrt{2}) \]

Final Calculation & Result:

Substitute \( I \) into the full expression \( E = I - 3 \log(\sqrt{3}) \), noting \( 3 \log(\sqrt{3}) = \frac{3}{2}\log(3) \).

\[ E = \left( 2 - \sqrt{2} + \frac{3}{2}\log(3) - \log(1+\sqrt{2}) \right) - \frac{3}{2}\log(3) \] \[ E = 2 - \sqrt{2} - \log(1+\sqrt{2}) \]

The value of the expression is \( 2 - \sqrt{2} - \log \left( 1 + \sqrt{2} \right) \).