`intsqrt(x^(2)-8x+7)dx=`

Options

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

Answer

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 need to complete the square for the quadratic expression \(x^2 - 8x + 7\).

\[ x^2 - 8x + 7 = (x^2 - 8x + 16) - 16 + 7 = (x - 4)^2 - 9 \]

### Step 2: Rewrite the integral
Now we can rewrite the integral using the completed square:

\[ \int \sqrt{(x - 4)^2 - 9} \, dx \]

### Step 3: Use a trigonometric substitution
To solve the integral, we can use the substitution \(x - 4 = 3 \sec(\theta)\). Then, \(dx = 3 \sec(\theta) \tan(\theta) \, d\theta\).

### Step 4: Substitute and simplify
Substituting into the integral gives:

\[ \int \sqrt{(3 \sec(\theta))^2 - 9} \cdot 3 \sec(\theta) \tan(\theta) \, d\theta \]

This simplifies to:

\[ \int \sqrt{9 \sec^2(\theta) - 9} \cdot 3 \sec(\theta) \tan(\theta) \, d\theta = \int \sqrt{9(\sec^2(\theta) - 1)} \cdot 3 \sec(\theta) \tan(\theta) \, d\theta \]

Using the identity \(\sec^2(\theta) - 1 = \tan^2(\theta)\):

\[ = \int \sqrt{9 \tan^2(\theta)} \cdot 3 \sec(\theta) \tan(\theta) \, d\theta = \int 3 \tan(\theta) \cdot 3 \sec(\theta) \tan(\theta) \, d\theta = 9 \int \tan^2(\theta) \sec(\theta) \, d\theta \]

### Step 5: Use the identity for \(\tan^2(\theta)\)
We know that \(\tan^2(\theta) = \sec^2(\theta) - 1\), so we can rewrite the integral:

\[ 9 \int (\sec^2(\theta) - 1) \sec(\theta) \, d\theta = 9 \left( \int \sec^3(\theta) \, d\theta - \int \sec(\theta) \, d\theta \right) \]

### Step 6: Integrate
The integral of \(\sec^3(\theta)\) can be solved using integration by parts, and the integral of \(\sec(\theta)\) is known:

\[ \int \sec(\theta) \, d\theta = \ln | \sec(\theta) + \tan(\theta) | + C \]

### Step 7: Back substitute
After calculating the integrals, we will substitute back in terms of \(x\) using the original substitution \(x - 4 = 3 \sec(\theta)\).

### Final Result
The final result will be in terms of \(x\) and will include a constant of integration \(C\).