Assertion (A) : `int(dx)/(x^(2) + 2x + 3) = (1)/(sqrt(2)) tan^(-1)((x+1)/(sqrt(2))) + c`
Reason (R) : `int(dx)/(x^(2) + a^(2)) = (1)/(a) tan^(-1)((x)/(a)) + c`

Options

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true and R is NOT the correct explanation of A.
C. A is true but R is false
D. A is false and R is true

Answer

To solve the integral \(\int \frac{dx}{x^2 + 2x + 3}\), we will first rewrite the quadratic expression in the denominator in a more manageable form.

### Step 1: Complete the square
The expression \(x^2 + 2x + 3\) can be rewritten by completing the square:
\[ x^2 + 2x + 3 = (x^2 + 2x + 1) + 2 = (x + 1)^2 + 2 \]
Thus, we have:
\[ \int \frac{dx}{x^2 + 2x + 3} = \int \frac{dx}{(x + 1)^2 + 2} \]

### Step 2: Use the formula for integration
We can use the formula for the integral of the form \(\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C\). In our case, we identify \(a^2 = 2\), so \(a = \sqrt{2}\).

### Step 3: Substitute into the formula
Now, we can apply the formula:
\[ \int \frac{dx}{(x + 1)^2 + (\sqrt{2})^2} = \frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x + 1}{\sqrt{2}}\right) + C \]

### Final Result
Thus, the integral evaluates to:
\[ \int \frac{dx}{x^2 + 2x + 3} = \frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x + 1}{\sqrt{2}}\right) + C \]

### Conclusion
This confirms the assertion (A) is correct. The reason (R) provided is also valid as it is a general formula for integrals of the form \(\int \frac{dx}{x^2 + a^2}\).