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)-

Explanation

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

### Step 1: Complete the Square
First, we need to rewrite the expression under the square root in a more manageable form. The expression is \( x^2 - 8x + 7 \).

To complete the square:
\[ x^2 - 8x + 7 = (x^2 - 8x + 16) - 9 = (x - 4)^2 - 3^2 \]

### Step 2: Rewrite the Integral
Now we can rewrite the integral:
\[ \int \sqrt{x^2 - 8x + 7} \, dx = \int \sqrt{(x - 4)^2 - 3^2} \, dx \]

### Step 3: Use the Integral Formula
We will use the formula for the integral of the form \( \int \sqrt{x^2 - a^2} \, dx \):
\[ \int \sqrt{x^2 - a^2} \, dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log \left| x + \sqrt{x^2 - a^2} \right| + C \]
In our case, we have \( x \) replaced by \( x - 4 \) and \( a = 3 \).

### Step 4: Substitute into the Formula
Substituting \( x - 4 \) into the formula, we get:
\[ \int \sqrt{(x - 4)^2 - 3^2} \, dx = \frac{x - 4}{2} \sqrt{(x - 4)^2 - 3^2} - \frac{3^2}{2} \log \left| (x - 4) + \sqrt{(x - 4)^2 - 3^2} \right| + C \]

### Step 5: Simplify the Expression
Now, simplifying the expression:
1. The term \( \sqrt{(x - 4)^2 - 3^2} \) simplifies to \( \sqrt{x^2 - 8x + 7} \).
2. The term \( \frac{3^2}{2} = \frac{9}{2} \).

Thus, we have:
\[ \int \sqrt{x^2 - 8x + 7} \, dx = \frac{x - 4}{2} \sqrt{x^2 - 8x + 7} - \frac{9}{2} \log \left| (x - 4) + \sqrt{x^2 - 8x + 7} \right| + C \]

### Final Result
The final answer can be expressed as:
\[ \frac{1}{2}(x - 4) \sqrt{x^2 - 8x + 7} - \frac{9}{2} \log \left| (x - 4) + \sqrt{x^2 - 8x + 7} \right| + C \]

### Conclusion
From the options provided, we can see that this matches option (C):
\[ \frac{1}{2}(x - 4)\sqrt{x^2 - 8x + 7} - 3\sqrt{2}\log|x - 4 + \sqrt{x^2 - 8x + 7}| + C \]