NVAMediumJEE Main 2025 · 24 January, Shift 2Indefinite Integrals

Mathematics Question from JEE Main 2025 · 24 January, Shift 2

If 2x2+5x+9x2+x+1dx=xx2+x+1+αx2+x+1+βloge(x+12+x2+x+1)+C,\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\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 CC is the constant of integration, then α+2β\alpha + 2\beta is equal to _____

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