Integrate the function: (4 x+2) x^ 2 +x+1 - Vidyadip

Question

Integrate the function: $(4 x+2) \sqrt{x^{2}+x+1}$

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Answer

Let $x^2 + x + 1 = t$
Differentiating both sides, we get,
$\Rightarrow (2x + 1)dx = dt$
Therefore
$\Rightarrow \int(4 x+2) \sqrt{x^{2}+x+1} d x=\int 2 \sqrt{t} d t$
$=2 \int \sqrt{t} d t$
$= 2\left(\frac{t^{\frac{3}{2}}}{\frac{3}{2}}\right)+C$
$=\frac{4}{3}\left(x^{2}+x+1\right)^{\frac{3}{2}}+C$

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