Choose the correct answer`intsqrt(x^2-8x+7)dx `(A)` `1/2(x-4)sqrt(x^2-8x+7)+9log|x-4+sqrt(x^2-8x+7)|+C` (B) `1/2(x+4)sqrt(x^2-8x+7)+9log|x+4+sqrt(x^2-8x+7)|+C`(C) `1/2(x-4)sqrt(x^2-8x+7)-3sqrt(2)log|x-4+sqrt(x^2-8x+7)|+C`(D)`1/2(x-4)sqrt(x^2-8x+7)-(9/2)log|x-4+sqrt(x^2-8x+7)|+C`

Options

A. `1/2(x-4)sqrt(x^2-8x+7)+9log|x-4+sqrt(x^2-8x+7)|+C`
B. `1/2(x+4)sqrt(x^2-8x+7)+9log|x+4+sqrt(x^2-8x+7)|+C`
C. `1/2(x-4)sqrt(x^2-8x+7)-3sqrt(2)log|x-4+sqrt(x^2-8x+7)|+C`
D. `1/2(x-4)sqrt(x^2-8x+7)-(9/2)log|x-4+sqrt(x^2-8x+7)|+C`

Explanation

To solve the integral \( \int \sqrt{x^2 - 8x + 7} \, dx \), we will follow these steps:

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

First, we rewrite the expression inside the square root:
\[ x^2 - 8x + 7 = (x^2 - 8x + 16) - 9 = (x - 4)^2 - 3^2 \]
Thus, we have:
\[ \sqrt{x^2 - 8x + 7} = \sqrt{(x - 4)^2 - 3^2} \]

### Step 2: Substitute for simplification

Let \( t = x - 4 \). Then, \( dx = dt \). The integral becomes:
\[ \int \sqrt{t^2 - 3^2} \, dt \]

### Step 3: Apply the integration formula

We use the formula for the integral of the form \( \int \sqrt{x^2 - a^2} \, dx \):
\[ \int \sqrt{x^2 - a^2} \, dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log |x + \sqrt{x^2 - a^2}| + C \]
In our case, \( a = 3 \). Thus, we apply the formula:
\[ \int \sqrt{t^2 - 3^2} \, dt = \frac{t}{2} \sqrt{t^2 - 9} - \frac{9}{2} \log |t + \sqrt{t^2 - 9}| + C \]

### Step 4: Substitute back in terms of \( x \)

Now we substitute back \( t = x - 4 \):
\[ = \frac{x - 4}{2} \sqrt{(x - 4)^2 - 9} - \frac{9}{2} \log |(x - 4) + \sqrt{(x - 4)^2 - 9}| + C \]

### Step 5: Simplify the square root

We simplify \( \sqrt{(x - 4)^2 - 9} \):
\[ \sqrt{(x - 4)^2 - 9} = \sqrt{x^2 - 8x + 7 - 9} = \sqrt{x^2 - 8x - 2} \]

### Final Answer

Thus, the integral evaluates to:
\[ \frac{1}{2}(x - 4) \sqrt{x^2 - 8x + 7} - \frac{9}{2} \log |x - 4 + \sqrt{x^2 - 8x + 7}| + C \]

Comparing with the options provided, we find that the correct answer is:
**(D)** \( \frac{1}{2}(x - 4)\sqrt{x^2 - 8x + 7} - \frac{9}{2}\log|x - 4 + \sqrt{x^2 - 8x + 7}| + C \)