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

Question

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

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Answer

$I=\int \sqrt{1-4 x-x^{2}} d x$
= $\int \sqrt{1-\left(x^{2}+4 x+4-4\right)} d x$
= $\int \sqrt{1+4-(x+2)^{2}} d x$
= $\int \sqrt{(\sqrt{5})^{2}-(x+2)^{2}} d x$
We know that,
$\Rightarrow \int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+C$
$\Rightarrow I=\frac{(\mathrm{x}+2)}{2} \sqrt{1-4 \mathrm{x}-\mathrm{x}^{2}}+\frac{5}{2} \sin ^{-1}\left(\frac{\mathrm{x}+2}{\sqrt{5}}\right)+\mathrm{C}$

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