Evaluate the following integrals: (4x+2) x^2+x+1 dx - Vidyadip

Question

Evaluate the following integrals:
$\int(4\text{x}+2)\sqrt{\text{x}^2+\text{x}+1}\text{ dx}$

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Answer

$\int(4\text{x}+2)\sqrt{\text{x}^2+\text{x}+1}\text{ dx}$
$=2\int(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}\text{ dx}$
$\text{Let }\text{x}^2+\text{x}+1=\text{t}$
$\Rightarrow(2\text{x}+1)=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow(2\text{x}+1)\text{dx}=\text{dt}$
$\text{Now, }2\int(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}\text{ dx}$
$=2\int\sqrt{\text{t}}\text{ dt}$
$=2\int\text{t}^\frac{1}{2}\text{dt}$
$=2\bigg[\frac{\text{t}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\bigg]+\text{C}$
$=2\times\frac{2}{3}\text{t}^\frac{3}{2}+\text{C}$
$=\frac{4}{3}\text{t}^\frac{3}{2}+\text{C}$
$=\frac{4}{3}(\text{x}^2+\text{x}+1)^\frac{3}{2}+\text{C}$

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