`intsqrt(5-2x+x^(2))dx`

Explanation

To solve the integral \(\int \sqrt{5 - 2x + x^2} \, dx\), we will follow a systematic approach. Let's break it down step by step.

### Step 1: Rewrite the Expression
First, we need to rewrite the expression under the square root in a more manageable form. The expression \(5 - 2x + x^2\) can be rearranged as follows:

\[ x^2 - 2x + 5 = (x^2 - 2x + 1) + 4 = (x - 1)^2 + 4 \]

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

\[ \int \sqrt{(x - 1)^2 + 4} \, dx \]

### Step 3: Use the Integral Formula
We can use the formula for the integral of the form \(\int \sqrt{x^2 + a^2} \, dx\), which is:

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

In our case, we have \(x\) as \(x - 1\) and \(a = 2\). Therefore, we will apply the formula accordingly.

### Step 4: Apply the Formula
Substituting \(x - 1\) for \(x\) in the formula:

\[ \int \sqrt{(x - 1)^2 + 4} \, dx = \frac{x - 1}{2} \sqrt{(x - 1)^2 + 4} + \frac{4}{2} \ln\left| (x - 1) + \sqrt{(x - 1)^2 + 4} \right| + C \]

### Step 5: Simplify the Expression
Now, simplifying the expression:

\[ = \frac{x - 1}{2} \sqrt{(x - 1)^2 + 4} + 2 \ln\left| (x - 1) + \sqrt{(x - 1)^2 + 4} \right| + C \]

### Final Result
Thus, the final result for the integral \(\int \sqrt{5 - 2x + x^2} \, dx\) is:

\[ \frac{x - 1}{2} \sqrt{(x - 1)^2 + 4} + 2 \ln\left| (x - 1) + \sqrt{(x - 1)^2 + 4} \right| + C \]