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

Answer

To solve the integral \(\int \sqrt{x^2 + 5} \, dx\), we can use a standard formula for integrating expressions of the form \(\sqrt{x^2 + a^2}\).

### Step-by-Step Solution:

1. **Identify the integral**: We have the integral \(\int \sqrt{x^2 + 5} \, dx\). Here, we can recognize that \(a^2 = 5\), which means \(a = \sqrt{5}\).

2. **Use the formula**: The formula for integrating \(\sqrt{x^2 + a^2}\) is:
\[ \int \sqrt{x^2 + a^2} \, dx = \frac{x}{2} \sqrt{x^2 + a^2} + \frac{a^2}{2} \ln \left| x + \sqrt{x^2 + a^2} \right| + C \]
where \(C\) is the constant of integration.

3. **Substitute \(a\)**: In our case, since \(a^2 = 5\), we have \(a = \sqrt{5}\). Now we can substitute \(a\) into the formula:
\[ \int \sqrt{x^2 + 5} \, dx = \frac{x}{2} \sqrt{x^2 + 5} + \frac{5}{2} \ln \left| x + \sqrt{x^2 + 5} \right| + C \]

4. **Final answer**: Thus, the integral evaluates to:
\[ \int \sqrt{x^2 + 5} \, dx = \frac{x}{2} \sqrt{x^2 + 5} + \frac{5}{2} \ln \left| x + \sqrt{x^2 + 5} \right| + C \]