Integrate the functions in Exercises: (4x+2) x^2+x+1 - Vidyadip

Question

Integrate the functions in Exercises:
$(4\text{x}+2)\sqrt{\text{x}^2+\text{x}+1}$

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Answer

$\text{Let I}=\int(4\text{x}+ 2)\sqrt{\text{x}^2+\text{x} +1}\text{ dx}$
$=\int2(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}\text{ dx}$
$=\int2\sqrt{\text{x}^2+\text{x}+1}( 2\text{x}+1)\text{ dx} \ \ \ \ \ \ \ ...\text{(i)} $
Putting $\text{ x}^2+\text{x}+1=\text{t} \ \ \ \Rightarrow \ \ \ (2\text{x + 1})=\frac{\text{dt}}{\text{dx}} \ \ \Rightarrow \ \ \ (2\text{x + 1})\text{ dx = dt}$
$\therefore\ \ \ \ \ $From eq. (i), $\text{I}=\int2\sqrt{\text{t}}\text{ dt}=2\int\text{t}^{\frac{1}{2}}\text{ dt}=2\frac{\text{t}^{^3/_2}}{^3/_2}+\text{c}=\frac{4}{3}\text{t}^{^3/_2}+\text{c} $
$=\frac{4}{3}\big(\text{x}^2+\text{x}+1\big)^{^3/_2}+\text{c} $

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