Choose the correct answer`int_1^sqrt(3) (dx)/(1+x^2)`equals(A) `pi/3` (B) `(2pi)/3` (C) `pi/6` (D) `pi/(12)`

Explanation

To solve the integral \(\int_1^{\sqrt{3}} \frac{dx}{1+x^2}\), we will follow these steps:

### Step 1: Identify the Integral
The integral we need to solve is:
\[ \int_1^{\sqrt{3}} \frac{dx}{1+x^2} \]

### Step 2: Use the Standard Integral Formula
We know that:
\[ \int \frac{dx}{1+x^2} = \tan^{-1}(x) + C \]
where \(C\) is the constant of integration.

### Step 3: Apply the Limits of Integration
Now we will apply the limits from 1 to \(\sqrt{3}\):
\[ \left[ \tan^{-1}(x) \right]_1^{\sqrt{3}} = \tan^{-1}(\sqrt{3}) - \tan^{-1}(1) \]

### Step 4: Evaluate the Inverse Tangent Values
Using known values of the inverse tangent:
- \(\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}\)
- \(\tan^{-1}(1) = \frac{\pi}{4}\)

Substituting these values into the expression:
\[ \frac{\pi}{3} - \frac{\pi}{4} \]

### Step 5: Find a Common Denominator
To subtract these fractions, we need a common denominator. The least common multiple of 3 and 4 is 12:
\[ \frac{\pi}{3} = \frac{4\pi}{12}, \quad \frac{\pi}{4} = \frac{3\pi}{12} \]

### Step 6: Perform the Subtraction
Now, we can subtract:
\[ \frac{4\pi}{12} - \frac{3\pi}{12} = \frac{1\pi}{12} = \frac{\pi}{12} \]

### Conclusion
Thus, the value of the integral is:
\[ \int_1^{\sqrt{3}} \frac{dx}{1+x^2} = \frac{\pi}{12} \]

The correct answer is option (D) \(\frac{\pi}{12}\).

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