### Related Formula Standard integration templates for quadratic forms: int sqrtt^2 + k^2 dt = fract2sqrtt^2+k^2 + frack^22lnleft|t + sqrtt^2+k^2right|$\int \sqrt{t^2 + k^2} dt = \frac{t}{2}\sqrt{t^2+k^2} + \frac{k^2}{2}\ln\left|t + \sqrt{t^2+k^2}\right|$ int frac1sqrtt^2 + k^2 dt = lnleft|t + sqrtt^2+k^2right|$\int \frac{1}{\sqrt{t^2 + k^2}} dt = \ln\left|t + \sqrt{t^2+k^2}\right|$ ### Core Logic Decompose the numerator using polynomial differentiation components : 2x^2 + 5x + 9 = A(x^2 + x + 1) + B(2x + 1) + C$2x^2 + 5x + 9 = A(x^2 + x + 1) + B(2x + 1) + C$ Equating coefficients dynamically yields : - For x^2$x^2$: A = 2$A = 2$ . - For x$x$: A + 2B = 5 Rightarrow 2 + 2B = 5 Rightarrow B = frac32$A + 2B = 5 \Rightarrow 2 + 2B = 5 \Rightarrow B = \frac{3}{2}$ . - Constant: A + B + C = 9 Rightarrow 2 + frac32 + C = 9 Rightarrow C = frac112$A + B + C = 9 \Rightarrow 2 + \frac{3}{2} + C = 9 \Rightarrow C = \frac{11}{2}$ . Rewrite the integrand into three parts : 2int sqrtx^2+x+1 dx + frac32int frac2x+1sqrtx^2+x+1 dx + frac112int frac1sqrtx^2+x+1 dx$2\int \sqrt{x^2+x+1} dx + \frac{3}{2}\int \frac{2x+1}{\sqrt{x^2+x+1}} dx + \frac{11}{2}\int \frac{1}{\sqrt{x^2+x+1}} dx$ ### Step 1: Complete Quadratics & Integrate Format using completing the square technique: x^2 + x + 1 = left(x + frac12right)^2 + left(fracsqrt32right)^2$x^2 + x + 1 = \left(x + \frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2$. 1. First Integral evaluation: 2 left[ fracleft(x+frac12right)2sqrtx^2+x+1 + frac38lnleft|x+frac12+sqrtx^2+x+1right| right]$2 \left[ \frac{\left(x+\frac{1}{2}\right)}{2}\sqrt{x^2+x+1} + \frac{3}{8}\ln\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right| \right]$ = left(x+frac12right)sqrtx^2+x+1 + frac34lnleft|x+frac12+sqrtx^2+x+1right|$= \left(x+\frac{1}{2}\right)\sqrt{x^2+x+1} + \frac{3}{4}\ln\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|$ 2. Second Integral evaluation : frac32 cdot 2sqrtx^2+x+1 = 3sqrtx^2+x+1$\frac{3}{2} \cdot 2\sqrt{x^2+x+1} = 3\sqrt{x^2+x+1}$ 3. Third Integral evaluation : frac112lnleft|x+frac12+sqrtx^2+x+1right|$\frac{11}{2}\ln\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|$ ### Step 2: Collect Like Terms Gather and combine matching factor parameters: textTotal = left(x + frac12 + 3right)sqrtx^2+x+1 + left(frac34 + frac112right)lnleft|x+frac12+sqrtx^2+x+1right|$\text{Total} = \left(x + \frac{1}{2} + 3\right)\sqrt{x^2+x+1} + \left(\frac{3}{4} + \frac{11}{2}\right)\ln\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|$ textTotal = left(x + frac72right)sqrtx^2+x+1 + frac254lnleft|x+frac12+sqrtx^2+x+1right|$\text{Total} = \left(x + \frac{7}{2}\right)\sqrt{x^2+x+1} + \frac{25}{4}\ln\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|$ = xsqrtx^2+x+1 + frac72sqrtx^2+x+1 + frac254lnleft|x+frac12+sqrtx^2+x+1right|$= x\sqrt{x^2+x+1} + \frac{7}{2}\sqrt{x^2+x+1} + \frac{25}{4}\ln\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|$ ### Step 3: Extract Coefficients Compare directly against the given expression variables : alpha = frac72, quad beta = frac254$\alpha = \frac{7}{2}, \quad \beta = \frac{25}{4}$ alpha + 2beta = frac72 + 2left(frac254right) = frac72 + frac252 = frac322 = 16$\alpha + 2\beta = \frac{7}{2} + 2\left(\frac{25}{4}\right) = \frac{7}{2} + \frac{25}{2} = \frac{32}{2} = 16$ ### Pattern Recognition When dividing large numerators containing x^2$x^2$ elements over quadratic square roots, using matching coefficient expansion rules prevents lengthy substitution errors completely. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Integrals