Question

If \[ \int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = \sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e \left( \left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| \right) + C, \] where \( C \) is the constant of integration, then \( \alpha + 2\beta \) is equal to …….. 

Hint:

To solve integrals involving quadratic expressions under square roots, try substitution methods to simplify the expression. Then, match the terms with the given result to find the unknown coefficients.

Answer 1

We are tasked with finding the values of \( \alpha \) and \( \beta \) in the given integral. To solve this, we perform the integration of the function \( \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \) using substitution and matching the result with the given expression. 
First, simplify the integrand by performing a substitution for \( u = x^2 + x + 1 \). This leads to a simpler form for the integral. We integrate and match the terms with the given solution. After performing the integration and comparing coefficients, we find that \( \alpha = 1 \) and \( \beta = -1 \). Thus, \[ \alpha + 2\beta = 1 + 2(-1) = 0. \] 
Final Answer: \( \alpha + 2\beta = 0 \).