If `I=int(dx)/(x^(2)sqrt(1+x^(2)))=(f(x))/(x)+C, (AA x gt0)` (where, C is the constant of integration) and `f(1)=-sqrt2`, then the value of `|f(sqrt3)|` is

Answer

To solve the given problem step by step, we start with the integral:

\[ I = \int \frac{dx}{x^2 \sqrt{1+x^2}} = \frac{f(x)}{x} + C \]

where \( C \) is the constant of integration and \( f(1) = -\sqrt{2} \). We need to find \( |f(\sqrt{3})| \).

### Step 1: Simplify the Integral
We begin by simplifying the integral:

\[ I = \int \frac{dx}{x^2 \sqrt{1+x^2}} \]

To simplify, we can factor out \( x^2 \) from the square root:

\[ I = \int \frac{dx}{x^2 \sqrt{x^2(1 + \frac{1}{x^2})}} = \int \frac{dx}{x^2 \cdot x \sqrt{1 + \frac{1}{x^2}}} = \int \frac{dx}{x^3 \sqrt{1 + \frac{1}{x^2}}} \]

### Step 2: Substitution
Next, we use substitution to make the integral easier to solve. Let:

\[ t = 1 + \frac{1}{x^2} \]

Then, differentiating \( t \):

\[ dt = -\frac{2}{x^3} dx \quad \Rightarrow \quad dx = -\frac{x^3}{2} dt \]

Substituting \( dx \) back into the integral gives:

\[ I = \int \frac{-\frac{x^3}{2} dt}{x^3 \sqrt{t}} = -\frac{1}{2} \int \frac{dt}{\sqrt{t}} \]

### Step 3: Integrate
Now we can integrate:

\[ -\frac{1}{2} \int t^{-\frac{1}{2}} dt = -\frac{1}{2} \cdot 2 t^{\frac{1}{2}} + C = -\sqrt{t} + C \]

Substituting back for \( t \):

\[ I = -\sqrt{1 + \frac{1}{x^2}} + C \]

### Step 4: Find \( f(x) \)
From our original equation, we have:

\[ \frac{f(x)}{x} = -\sqrt{1 + \frac{1}{x^2}} + C \]

Multiplying through by \( x \):

\[ f(x) = -x \sqrt{1 + \frac{1}{x^2}} + Cx \]

This simplifies to:

\[ f(x) = -\sqrt{x^2 + 1} + Cx \]

### Step 5: Use Initial Condition
Now we use the condition \( f(1) = -\sqrt{2} \):

\[ f(1) = -\sqrt{1^2 + 1} + C \cdot 1 = -\sqrt{2} + C \]

Setting this equal to \( -\sqrt{2} \):

\[ -\sqrt{2} + C = -\sqrt{2} \quad \Rightarrow \quad C = 0 \]

Thus, we have:

\[ f(x) = -\sqrt{x^2 + 1} \]

### Step 6: Find \( |f(\sqrt{3})| \)
Now we need to find \( |f(\sqrt{3})| \):

\[ f(\sqrt{3}) = -\sqrt{(\sqrt{3})^2 + 1} = -\sqrt{3 + 1} = -\sqrt{4} = -2 \]

Taking the modulus:

\[ |f(\sqrt{3})| = |-2| = 2 \]

### Final Answer
Thus, the value of \( |f(\sqrt{3})| \) is:

\[ \boxed{2} \]