`int sqrt(x^2-8x+7)` is equal to

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)-3sqrt2log |(x-4)+sqrt(x^2-8x+7)|+c`
D. `1/2(x-4)sqrt(x^2-8x+7)-9/2log |(x-4)+sqrt(x^2-8x+7)|+c`

Answer

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

### Step 1: Simplify the expression under the square root
First, we need to simplify the expression inside the square root:
\[ x^2 - 8x + 7 = (x^2 - 8x + 16) - 9 = (x - 4)^2 - 3^2 \]
Thus, we can rewrite the integral as:
\[ I = \int \sqrt{(x - 4)^2 - 3^2} \, dx \]

### Step 2: Use the integral formula
We can 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, \( a = 3 \) and \( x \) is replaced by \( x - 4 \). Therefore, we can apply the formula:
\[ I = \frac{x - 4}{2} \sqrt{(x - 4)^2 - 3^2} - \frac{3^2}{2} \log \left| (x - 4) + \sqrt{(x - 4)^2 - 3^2} \right| + C \]

### Step 3: Simplify the expression
Now, we simplify the expression:
\[ I = \frac{x - 4}{2} \sqrt{(x - 4)^2 - 9} - \frac{9}{2} \log \left| (x - 4) + \sqrt{(x - 4)^2 - 9} \right| + C \]

### Final Answer
Thus, the final result of the integral is:
\[ I = \frac{x - 4}{2} \sqrt{(x - 4)^2 - 9} - \frac{9}{2} \log \left| (x - 4) + \sqrt{(x - 4)^2 - 9} \right| + C \]

---