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 ……..
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 ……..
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Solution and Explanation
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: \( \alpha + 2\beta = 0 \).
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