JEE Main 2025 — 24 January, Evening Shift — Question 23

$2 \mathrm{x}^{2}+5 \mathrm{x}+9=\mathrm{A}\left(\mathrm{x}^{2}+\mathrm{x}+1\right)+\mathrm{B}(2 \mathrm{x}+1)+\mathrm{C}$

$A=2$, $B=\frac{3}{2}$ ,$C=\frac{11}{2}$

$2 \int \sqrt{x^{2}+x+1} d x+\frac{3}{2} \int \frac{2 x+1}{\sqrt{x^{2}+x+1}} d x+\frac{11}{2} \int \frac{d x}{\sqrt{x^{2}+x+1}}$

$2 \int \sqrt{\left(x+\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}} d x+3 \sqrt{x^{2}+x+1}+\frac{11}{2} \int \frac{d x}{\sqrt{\left(x+\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}}$

$2\left(\frac{x+\frac{1}{2}}{2} \sqrt{x^{2}+x+1}+\frac{3}{8} \ln \left(x+\frac{1}{2}+\sqrt{x^{2}+x+1}\right)\right)+3 \sqrt{x^{2}+x+1}$

$+\frac{11}{2} \ln \left(x+\frac{1}{2}+\sqrt{x^{2}+x+1}\right)+C$

$\alpha=\frac{7}{2}, \quad \beta=\frac{25}{4}$

$\alpha+2 \beta=16$