If `int(x)/(x^(2)-4x+8)dx=Klog(x^(2)-4x+8)+tan^(-1)((x-2)/(2)) + C` then the value of K is

Options

A. 1\2
B. 1
C. 2
D. none of these

Answer

To find the value of \( K \) in the given integral equation:

\[ \int \frac{x}{x^2 - 4x + 8} \, dx = K \log(x^2 - 4x + 8) + \tan^{-1}\left(\frac{x - 2}{2}\right) + C \]

we will solve the integral step by step.

### Step 1: Rewrite the Integral
We start with the integral:

\[ I = \int \frac{x}{x^2 - 4x + 8} \, dx \]

### Step 2: Simplify the Denominator
First, we can differentiate the denominator \( x^2 - 4x + 8 \):

\[ \frac{d}{dx}(x^2 - 4x + 8) = 2x - 4 \]

We can express \( x \) in terms of \( 2x - 4 \) to facilitate integration.

### Step 3: Split the Integral
We can rewrite \( x \) as follows:

\[ x = \frac{1}{2}(2x - 4) + 2 \]

Thus, we can split the integral:

\[ I = \int \frac{1}{2}(2x - 4) \cdot \frac{1}{x^2 - 4x + 8} \, dx + \int \frac{2}{x^2 - 4x + 8} \, dx \]

### Step 4: Change of Variable for the First Integral
Let \( t = x^2 - 4x + 8 \), then \( dt = (2x - 4) \, dx \). Therefore, the first integral becomes:

\[ \int \frac{dt}{2t} = \frac{1}{2} \log |t| + C_1 = \frac{1}{2} \log |x^2 - 4x + 8| + C_1 \]

### Step 5: Evaluate the Second Integral
Now we evaluate the second integral:

\[ \int \frac{2}{x^2 - 4x + 8} \, dx \]

We can complete the square in the denominator:

\[ x^2 - 4x + 8 = (x - 2)^2 + 4 \]

Thus, we have:

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

This integral can be solved using the formula:

\[ \int \frac{a}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \]

Here, \( a = 2 \):

\[ = \tan^{-1}\left(\frac{x - 2}{2}\right) + C_2 \]

### Step 6: Combine the Results
Combining both parts of the integral, we have:

\[ I = \frac{1}{2} \log(x^2 - 4x + 8) + \tan^{-1}\left(\frac{x - 2}{2}\right) + C \]

### Step 7: Compare with the Given Expression
Now, comparing this with the given expression:

\[ K \log(x^2 - 4x + 8) + \tan^{-1}\left(\frac{x - 2}{2}\right) + C \]

We can see that \( K = \frac{1}{2} \).

### Final Answer
Thus, the value of \( K \) is:

\[ \boxed{\frac{1}{2}} \]