Integrate the function x^ 2 +4 x+1 - Vidyadip

Question

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

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Answer

Let $I=\int \sqrt{x^{2}+4 x+1} d x$
$=\int \sqrt{\left(x^{2}+4 x+4\right)-3} d x$
$= \int \sqrt{(x+2)^{2}-(\sqrt{3})^{2}} d x$
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,
$\Rightarrow \mathrm{I}=\frac{(\mathrm{x}+2)}{2} \sqrt{\mathrm{x}^{2}+4 \mathrm{x}+1}-\frac{3}{2} \log |(\mathrm{x}+2)+\sqrt{\mathrm{x}^{2}+4 \mathrm{x}+1}|+\mathrm{C}$

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