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Integrate the function
$(4 x+2) \sqrt{x^2+x+1}$
$(4 x+2) \sqrt{x^2+x+1}$
Solution:
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Verified Answer
Let $x^2+x+1=t \quad \Rightarrow \quad(2 x+1) d x=d t$
$\begin{aligned}
&\therefore \quad \int(4 x+2) \sqrt{x^2+x+1} d x=2 \int \sqrt{t} d t \\
&=\frac{2 t^{3 / 2}}{3 / 2}+\mathrm{C}=\frac{4}{3} t^{3 / 2}+\mathrm{C} \\
&=\frac{4}{3}\left(x^2+x+1\right)^{3 / 2}+\mathrm{C}
\end{aligned}$
$\begin{aligned}
&\therefore \quad \int(4 x+2) \sqrt{x^2+x+1} d x=2 \int \sqrt{t} d t \\
&=\frac{2 t^{3 / 2}}{3 / 2}+\mathrm{C}=\frac{4}{3} t^{3 / 2}+\mathrm{C} \\
&=\frac{4}{3}\left(x^2+x+1\right)^{3 / 2}+\mathrm{C}
\end{aligned}$
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