Question
Evaluate the following integrals:
$\int\frac{\text{x}^3+\text{x}^2+2\text{x}+1}{\text{x}^2-\text{x}+1}\text{ dx}$
$\int\frac{\text{x}^3+\text{x}^2+2\text{x}+1}{\text{x}^2-\text{x}+1}\text{ dx}$
Comparing the coefficients of like powers of x,
$3=2\lambda\Rightarrow\lambda=\frac{3}{2}$ $-\lambda+\mu=-1\Rightarrow-\Big(\frac{3}{2}\Big)+\mu=-1$ $\mu=\frac{1}{2}$ So, $\text{I}_1=\int\frac{\frac{3}{2}(2\text{x}-1)+\frac{1}{2}}{\text{x}^2-\text{x}+1}\text{ dx}$ $\text{I}_1=\frac{3}{2}\int\frac{(2\text{x}-1)}{\text{x}^2-\text{x}+1}\text{ dx}+\frac{1}{2}\int\frac{1}{\text{x}^2-2\text{x}\big(\frac{1}{2}\big)+\big(\frac{1}{2}\big)^2-\big(\frac{1}{2}\big)^2+1}\text{ dx}$ $\text{I}_1=\frac{3}{2}\int\frac{2\text{x}-1}{\text{x}^2-\text{x}+1}\text{ dx}+\frac{1}{2}\int\frac{1}{\big(\text{x}-\frac{1}{2}\big)^2-\Big(\frac{\sqrt3}{2}\Big)^2}\text{ dx}$ $\text{I}_1=\frac{3}{2}\log\big|\text{x}^2-\text{x}+1\big|+\frac{1}{2}\times\frac{2}{\sqrt3}\tan^{-1}\bigg(\frac{\text{x}+\frac{1}{2}}{\frac{\sqrt3}{2}}\bigg)+\text{C}_2$ $\Big[\text{since},\int\frac{1}{\text{x}^2+\text{a}^2}\text{ dx}=\frac{1}{\text{a}}\tan^{-1}\Big(\frac{\text{x}}{\text{a}}\Big)+\text{C}\Big]$ $\text{I}_1=\frac{3}{2}\log\big|\text{x}^2-\text{x}+1\big|+\frac{1}{\sqrt3}\tan^{-1}\Big(\frac{2\text{x}+1}{\sqrt3}\Big)+\text{C}_2\ .....(2)$ Using equation (1) and (2) $\text{I}=\frac{\text{x}^2}{2}+2\text{x}+\frac{3}{2}\log\big|\text{x}^2-\text{x}+1\big|+\frac{1}{\sqrt3}\tan^{-1}\Big(\frac{2\text{x}+1}{\sqrt3}\Big)+\text{C}$Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
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