`int(1dx)/((x-2)sqrt(x^(2)-4x+3) `

Answer

To solve the integral

\[ \int \frac{1}{(x-2) \sqrt{x^2 - 4x + 3}} \, dx, \]

we can follow these steps:

### Step 1: Simplify the expression under the square root

First, we simplify the expression \(x^2 - 4x + 3\):

\[ x^2 - 4x + 3 = (x-2)^2 - 1. \]

This can be verified by expanding \((x-2)^2\):

\[ (x-2)^2 = x^2 - 4x + 4. \]

Thus,

\[ (x-2)^2 - 1 = x^2 - 4x + 3. \]

### Step 2: Rewrite the integral

Now, substituting this back into the integral, we have:

\[ \int \frac{1}{(x-2) \sqrt{(x-2)^2 - 1}} \, dx. \]

### Step 3: Make a substitution

Let \(t = x - 2\). Then, \(dx = dt\). The integral becomes:

\[ \int \frac{1}{t \sqrt{t^2 - 1}} \, dt. \]

### Step 4: Use the known integral formula

The integral

\[ \int \frac{1}{t \sqrt{t^2 - 1}} \, dt \]

is a standard integral that evaluates to:

\[ \sec^{-1} |t| + C. \]

### Step 5: Substitute back to the original variable

Now, substituting \(t = x - 2\) back into the expression, we get:

\[ \sec^{-1} |x - 2| + C. \]

### Final Answer

Thus, the final result of the integral is:

\[ \int \frac{1}{(x-2) \sqrt{x^2 - 4x + 3}} \, dx = \sec^{-1} |x - 2| + C. \]

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