Evaluate the following integrals: x^2+x+1 x^2+x+1 dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{\text{x}^2+\text{x}+1}{\text{x}^2+\text{x}+1}\text{dx}$

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Answer

Let $\text{I}=\int\frac{\text{x}^2+\text{x}+1}{\text{x}^2+\text{x}+1}\text{dx}$

$=\int\begin{bmatrix}1+\frac{\text{2x}}{\text{x}^2-\text{x}+1}\end{bmatrix}\text{dx}$

$\text{I}=\text{x}+\int\frac{2\text{x}}{\text{x}^2-\text{x}+1}\text{ dx}+\text{C}_1\ ....(1)$

Let $\text{I}=\int\frac{2\text{x}}{\text{x}^2-\text{x}+1}\text{ dx}$

Let $2\text{x}=\lambda\frac{\text{d}}{\text{dx}}\big(\text{x}^2-\text{x}+1\big)+\mu$

$=\lambda(2\text{x}-1)+\mu$

$2\text{x}=(2\lambda)\text{x}-\lambda+\mu$

Comparing the coefficients of like powers of x,

$2=2\lambda$

$\Rightarrow\lambda=1$

$-\lambda+\mu=0$

$\Rightarrow-1+\mu=0$

$\mu=1$

So, $\text{I}_1=\int\frac{(2\text{x}-1)+1}{\text{x}^2-\text{x}+1}\text{ dx}$

$=\int\frac{(2\text{x}-1)}{\text{x}^2-\text{x}+1}\text{ dx}+\int\frac{1}{\text{x}^2-2\text{x}\big(\frac{1}{2}\big)+\big(\frac{1}{2}\big)^2-\big(\frac{1}{2}\big)^2+1}\text{ dx}$

$\text{I}_1=\int\frac{2\text{x}-1}{\text{x}^2-\text{x}+1}\text{ dx+}\int\frac{1}{\big(\text{x}-\frac{1}{2}\big)^2-\Big(\frac{\sqrt3}{2}\Big)^2}\text{ dx}$

$=\log\big|\text{x}^2-\text{x}+1\big|+\frac{2}{\sqrt3}\tan^{-1}\bigg(\frac{\text{x}-\frac{1}{2}}{\frac{\sqrt3}{2}}\bigg)+\text{C}_2$

$\Big[\text{since},\int\frac{1}{\text{x}^2+\text{a}^2}\text{ dx}=\frac{1}{\text{a}}\tan^{-1}\Big(\frac{\text{x}}{\text{a}}\Big)+\text{C}\Big]$

$=\log\big|\text{x}^2-\text{x}+1\big|+\frac{2}{\sqrt3}\tan^{-1}\Big(\frac{2\text{x}-1}{\sqrt3}\Big)+\text{C}_2\ .....(2)$

Using equation (1) and (2)

$\text{I}=\text{x}+\log\big|\text{x}^2-\text{x}+1\big|+\frac{2}{\sqrt3}\tan^{-1}\Big(\frac{2\text{x}-1}{\sqrt3}\Big)+\text{C}$

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