Evaluate the following integrals:
`int sqrt(10-4x+x^(2))dx`

Answer

To evaluate the integral \( \int \sqrt{10 - 4x + x^2} \, dx \), we will first rewrite the expression under the square root in a more manageable form by completing the square.

### Step 1: Complete the square
The expression \( 10 - 4x + x^2 \) can be rearranged. We start by rewriting it as follows:

\[ x^2 - 4x + 10 \]

To complete the square for \( x^2 - 4x \), we take the coefficient of \( x \) (which is -4), halve it to get -2, and then square it to get 4. We can then rewrite the expression as:

\[ x^2 - 4x + 4 + 10 - 4 = (x - 2)^2 + 6 \]

Thus, we have:

\[ \sqrt{10 - 4x + x^2} = \sqrt{(x - 2)^2 + 6} \]

### Step 2: Substitute into the integral
Now we can substitute this back into the integral:

\[ \int \sqrt{10 - 4x + x^2} \, dx = \int \sqrt{(x - 2)^2 + 6} \, dx \]

### Step 3: Use the integral formula
We can use the formula for integrating \( \sqrt{x^2 + a^2} \):

\[ \int \sqrt{x^2 + a^2} \, dx = \frac{x}{2} \sqrt{x^2 + a^2} + \frac{a^2}{2} \ln |x + \sqrt{x^2 + a^2}| + C \]

In our case, we set \( x = x - 2 \) and \( a^2 = 6 \) (thus \( a = \sqrt{6} \)).

### Step 4: Apply the formula
Now we apply the formula:

1. Substitute \( x \) with \( (x - 2) \):
\[ \int \sqrt{(x - 2)^2 + 6} \, dx = \frac{(x - 2)}{2} \sqrt{(x - 2)^2 + 6} + \frac{6}{2} \ln |(x - 2) + \sqrt{(x - 2)^2 + 6}| + C \]

2. Simplify:
\[ = \frac{(x - 2)}{2} \sqrt{(x - 2)^2 + 6} + 3 \ln |(x - 2) + \sqrt{(x - 2)^2 + 6}| + C \]

### Step 5: Substitute back
Finally, we substitute back \( x - 2 \) to get the final answer in terms of \( x \):

\[ = \frac{(x - 2)}{2} \sqrt{10 - 4x + x^2} + 3 \ln |x - 2 + \sqrt{10 - 4x + x^2}| + C \]

### Final Answer
Thus, the evaluated integral is:

\[ \int \sqrt{10 - 4x + x^2} \, dx = \frac{(x - 2)}{2} \sqrt{10 - 4x + x^2} + 3 \ln |x - 2 + \sqrt{10 - 4x + x^2}| + C \]