from sympy import symbols, integrate

Define the variable and the function

x = symbols('x') expr = 1 / (2x**2 + 5x + 1)

Perform the integration

integral = integrate(expr, x) integralsqrt(17)*log(x - sqrt(17)/4 + 5/4)/17 - sqrt(17)*log(x + sqrt(17)/4 + 5/4)/17sqrt(17)*log(x - sqrt(17)/4 + 5/4)/17 - sqrt(17)*log(x + sqrt(17)/4 + 5/4)/17successThe solution to the given integral is:

$\int \frac{dx}{2x^2 + 5x + 1} = \frac{\sqrt{17}}{17} \ln\left(x - \frac{\sqrt{17}}{4} + \frac{5}{4}\right) - \frac{\sqrt{17}}{17} \ln\left(x + \frac{\sqrt{17}}{4} + \frac{5}{4}\right) + C$

Where $C$ is the constant of integration.

Let me know if you need further clarification or have more questions.

Here are five related questions you might explore:

1. How can this integral be solved using partial fraction decomposition?
2. What is the domain of the function in the integral?
3. How does completing the square help in solving such integrals?
4. Can we solve this integral numerically for a specific range of $x$?
5. How does the logarithmic function's behavior influence the solution?

Tip: Always check if completing the square simplifies the denominator in a quadratic integral before attempting partial fractions!