The solution of `t` the equation `int_(sqrt2)^(t) (dx)/(xsqrt((x^2 - 1))) = pi/12` is

Options

A. `2`
B. `sqrt(2)`
C. `2sqrt(3)`
D. `1`

Answer

To solve the equation

\[ \int_{\sqrt{2}}^{t} \frac{dx}{x \sqrt{x^2 - 1}} = \frac{\pi}{12} \]

we will follow these steps:

### Step 1: Identify the Integral
The integral

\[ \int \frac{dx}{x \sqrt{x^2 - 1}} \]

is known to be

\[ \sec^{-1}(x) + C \]

where \( C \) is the constant of integration.

### Step 2: Apply the Limits of Integration
Using the limits of integration from \(\sqrt{2}\) to \(t\), we can express the integral as:

\[ \sec^{-1}(t) - \sec^{-1}(\sqrt{2}) = \frac{\pi}{12} \]

### Step 3: Calculate \(\sec^{-1}(\sqrt{2})\)
We know that

\[ \sec(\frac{\pi}{4}) = \sqrt{2} \]

Thus,

\[ \sec^{-1}(\sqrt{2}) = \frac{\pi}{4} \]

### Step 4: Substitute and Rearrange
Substituting this value back into the equation gives:

\[ \sec^{-1}(t) - \frac{\pi}{4} = \frac{\pi}{12} \]

Adding \(\frac{\pi}{4}\) to both sides:

\[ \sec^{-1}(t) = \frac{\pi}{4} + \frac{\pi}{12} \]

### Step 5: Find a Common Denominator
To add these fractions, we find a common denominator. The least common multiple of 4 and 12 is 12:

\[ \frac{\pi}{4} = \frac{3\pi}{12} \]

Thus,

\[ \sec^{-1}(t) = \frac{3\pi}{12} + \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3} \]

### Step 6: Solve for \(t\)
Taking the secant of both sides:

\[ t = \sec\left(\frac{\pi}{3}\right) \]

We know that

\[ \sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2 \]

### Final Answer
Thus, the solution for \(t\) is:

\[ \boxed{2} \]