Question
Evaluate the following integrals:
$\int(\text{x}+2)\sqrt{\text{x}^2+\text{x}+1}\text{dx}$
$\int(\text{x}+2)\sqrt{\text{x}^2+\text{x}+1}\text{dx}$
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Answer
Let $\text{I}=\int(\text{x}+2)\sqrt{\text{x}^2+\text{x}+1}\text{dx}$
Let $\text{x}+2=\lambda\frac{\text{d}}{\text{dx}}(\text{x}^2+\text{x}+1)+\mu$
$=\lambda(2\text{x}+1)+\mu$
Equating similar terms, we get,
$2\lambda=1\Rightarrow\lambda=\frac{1}{2}$
$\lambda+\mu=2\Rightarrow\mu=2-\lambda=\frac{3}{2}$
$\therefore\ \mu=\frac{3}{2}$
$\therefore\ \text{I}=\int\Big(\frac{1}{2}(2\text{x}+1)+\frac{3}{2}\Big)\sqrt{\text{x}^2+\text{x}+1}\text{dx}$
$=\frac{1}{2}\int(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}+\frac{3}{2}\int\sqrt{\text{x}^2+\text{x}+1}\text{dx}$
Let $\text{x}^2+\text{x}+1=\text{t}$
$(2\text{x}+1)\text{dx = dt}$
$\therefore\ \text{I}=\frac{1}{2}\int\sqrt{\text{t}}\text{dt}+\frac{3}{2}\int\sqrt{\Big(\text{x}+\frac{1}{2}\Big)^2+\Big(\frac{\sqrt3}{2}\Big)^2}\text{dx}$
$\Rightarrow\text{I}=\frac{1}{2}\frac{\text{t}^{\frac{3}{2}}}{\frac{3}{2}}+\frac{3}{2}\begin{Bmatrix}\frac{\big(\text{x}+\frac{1}{2}\big)}{2}\sqrt{\text{x}^2+\text{x}+1}\\+\frac{3}{8}\log\Big|\Big(\text{x}+\frac{1}{2}\Big)+\sqrt{\text{x}^2+\text{x}+1}\Big|\end{Bmatrix}+\text{C}$
Hence,
$\Rightarrow\text{I}=\frac{1}{3}(\text{x}^2+\text{x}+1)^{\frac{3}{2}}+\frac{3}{8}(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}\\+\frac{9}{16}\log\Big|\Big(\text{x}+\frac{1}{2}\Big)+\sqrt{\text{x}^2+\text{x}+1}\Big|+\text{C}$
Let $\text{x}+2=\lambda\frac{\text{d}}{\text{dx}}(\text{x}^2+\text{x}+1)+\mu$
$=\lambda(2\text{x}+1)+\mu$
Equating similar terms, we get,
$2\lambda=1\Rightarrow\lambda=\frac{1}{2}$
$\lambda+\mu=2\Rightarrow\mu=2-\lambda=\frac{3}{2}$
$\therefore\ \mu=\frac{3}{2}$
$\therefore\ \text{I}=\int\Big(\frac{1}{2}(2\text{x}+1)+\frac{3}{2}\Big)\sqrt{\text{x}^2+\text{x}+1}\text{dx}$
$=\frac{1}{2}\int(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}+\frac{3}{2}\int\sqrt{\text{x}^2+\text{x}+1}\text{dx}$
Let $\text{x}^2+\text{x}+1=\text{t}$
$(2\text{x}+1)\text{dx = dt}$
$\therefore\ \text{I}=\frac{1}{2}\int\sqrt{\text{t}}\text{dt}+\frac{3}{2}\int\sqrt{\Big(\text{x}+\frac{1}{2}\Big)^2+\Big(\frac{\sqrt3}{2}\Big)^2}\text{dx}$
$\Rightarrow\text{I}=\frac{1}{2}\frac{\text{t}^{\frac{3}{2}}}{\frac{3}{2}}+\frac{3}{2}\begin{Bmatrix}\frac{\big(\text{x}+\frac{1}{2}\big)}{2}\sqrt{\text{x}^2+\text{x}+1}\\+\frac{3}{8}\log\Big|\Big(\text{x}+\frac{1}{2}\Big)+\sqrt{\text{x}^2+\text{x}+1}\Big|\end{Bmatrix}+\text{C}$
Hence,
$\Rightarrow\text{I}=\frac{1}{3}(\text{x}^2+\text{x}+1)^{\frac{3}{2}}+\frac{3}{8}(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}\\+\frac{9}{16}\log\Big|\Big(\text{x}+\frac{1}{2}\Big)+\sqrt{\text{x}^2+\text{x}+1}\Big|+\text{C}$
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