If `int _(-alpha) ^(alpha ) (e ^(x) + cos x ln (x + sqrt(1+x ^(2))))dx gt (3)/(2),` then posible value of `alpha` can be:

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to evaluate the integral and find the possible values of \( \alpha \) such that:

\[ \int_{-\alpha}^{\alpha} \left( e^x + \cos x \ln(x + \sqrt{1+x^2}) \right) dx > \frac{3}{2} \]

### Step 1: Define the Integral
Let:
\[ I = \int_{-\alpha}^{\alpha} \left( e^x + \cos x \ln(x + \sqrt{1+x^2}) \right) dx \]

### Step 2: Use the Property of Definite Integrals
We can use the property of definite integrals that states:
\[ \int_{-a}^{a} f(x) \, dx = \int_{0}^{a} (f(x) + f(-x)) \, dx \]
for an even function.

### Step 3: Analyze the Function
We need to check if the function \( f(x) = e^x + \cos x \ln(x + \sqrt{1+x^2}) \) is even or odd.

1. **Evaluate \( f(-x) \)**:
\[ f(-x) = e^{-x} + \cos(-x) \ln(-x + \sqrt{1+x^2}) = e^{-x} + \cos x \ln(-x + \sqrt{1+x^2}) \]

2. **Check if \( f(x) + f(-x) \) simplifies**:
\[ f(x) + f(-x) = e^x + e^{-x} + \cos x \left( \ln(x + \sqrt{1+x^2}) + \ln(-x + \sqrt{1+x^2}) \right) \]
The logarithmic terms can be simplified using the property of logarithms:
\[ \ln(x + \sqrt{1+x^2}) + \ln(-x + \sqrt{1+x^2}) = \ln\left((x + \sqrt{1+x^2})(-x + \sqrt{1+x^2})\right) \]
This simplifies to:
\[ \ln(1) = 0 \]
Hence:
\[ f(x) + f(-x) = e^x + e^{-x} = 2 \cosh(x) \]

### Step 4: Calculate the Integral
Thus, we can rewrite the integral:
\[ I = \int_{-\alpha}^{\alpha} f(x) \, dx = \int_{0}^{\alpha} 2 \cosh(x) \, dx \]
Calculating this integral:
\[ I = 2 \left[ \sinh(x) \right]_{0}^{\alpha} = 2 \left( \sinh(\alpha) - \sinh(0) \right) = 2 \sinh(\alpha) \]

### Step 5: Set up the Inequality
Now we have:
\[ 2 \sinh(\alpha) > \frac{3}{2} \]
Dividing both sides by 2:
\[ \sinh(\alpha) > \frac{3}{4} \]

### Step 6: Solve for \( \alpha \)
To find \( \alpha \), we take the inverse hyperbolic sine:
\[ \alpha > \sinh^{-1}\left(\frac{3}{4}\right) \]

### Step 7: Calculate \( \sinh^{-1}\left(\frac{3}{4}\right) \)
Using the formula for inverse hyperbolic sine:
\[ \sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1}) \]
Thus:
\[ \sinh^{-1}\left(\frac{3}{4}\right) = \ln\left(\frac{3}{4} + \sqrt{\left(\frac{3}{4}\right)^2 + 1}\right) = \ln\left(\frac{3}{4} + \sqrt{\frac{9}{16} + 1}\right) = \ln\left(\frac{3}{4} + \sqrt{\frac{25}{16}}\right) = \ln\left(\frac{3}{4} + \frac{5}{4}\right) = \ln(2) \]

### Conclusion
Therefore, the possible values of \( \alpha \) are:
\[ \alpha > \ln(2) \]