Integrate the function x^ 2 +4 x+6 - Vidyadip

Question

Integrate the function $\sqrt{x^{2}+4 x+6}$

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Answer

$I=\int \sqrt{x^{2}+4 x+6} d x$
= $\int \sqrt{x^{2}+4 x+4+2} d x$
= $\int \sqrt{(x+2)^{2}+(\sqrt{2})^{2}} d x$
We know that,
$\int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \log |x+\sqrt{x^{2}+a^{2}}|+C$
Therefore,
$I=\frac{(x+2)}{2} \sqrt{x^{2}+4 x+6}+\frac{2}{2} \log |(x+2)+\sqrt{x^{2}+4 x+6}|+C$
= $\frac{(x+2)}{2} \sqrt{x^{2}+4 x+6}+\log |(x+2)+\sqrt{x^{2}+4 x+6}|+C$

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