`int_(1)^(2)(1)/(x sqrt(x^(2)-1))dx=`

Options

A. `(pi)/(2)`
B. `(pi)/(3)`
C. `(pi)/(4)`
D. `(pi)/(6)`

Answer

To solve the integral \( I = \int_{1}^{2} \frac{1}{x \sqrt{x^2 - 1}} \, dx \), we will follow these steps:

### Step 1: Identify the Integral
We start with the integral:
\[ I = \int_{1}^{2} \frac{1}{x \sqrt{x^2 - 1}} \, dx \]

### Step 2: Use a Known Integral Formula
We recognize that the integral can be solved using the formula:
\[ \int \frac{1}{x \sqrt{x^2 - 1}} \, dx = \sec^{-1}(x) + C \]
Thus, we can rewrite our integral as:
\[ I = \left[ \sec^{-1}(x) \right]_{1}^{2} \]

### Step 3: Evaluate the Integral at the Limits
Next, we evaluate the integral at the upper and lower limits:
\[ I = \sec^{-1}(2) - \sec^{-1}(1) \]

### Step 4: Calculate the Values of the Inverse Secant
We know that:
- \(\sec^{-1}(2) = \frac{\pi}{3}\) (since \(\sec(\frac{\pi}{3}) = 2\))
- \(\sec^{-1}(1) = 0\) (since \(\sec(0) = 1\))

Substituting these values back into our expression for \(I\):
\[ I = \frac{\pi}{3} - 0 = \frac{\pi}{3} \]

### Step 5: Conclusion
Thus, the value of the definite integral is:
\[ I = \frac{\pi}{3} \]

### Final Answer
The answer is:
\[ \frac{\pi}{3} \]

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