Evaluate the following: 5-2x+x^2 dx - Vidyadip

Question

Evaluate the following:
$\int\sqrt{5-2\text{x}+\text{x}^2}\text{dx}$

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Answer

Let $\text{I}=\int\sqrt{5-2\text{x}+\text{x}^2}\text{dx}$
$=\int\sqrt{\text{x}^2-2\text{x}+1+4}\text{dx}$
$=\int\sqrt{(\text{x}-1)^2+(2)^2}\text{dx}$
Using $\int\sqrt{\text{x}^2+\text{a}^2}\text{dx}$ $=\frac{1}{2}\text{x}\sqrt{\text{x}^2+\text{a}^2}+\frac{\text{a}^2}{2}\log\Big|\text{x}+\sqrt{\text{x}^2+\text{a}^2}\Big|+\text{C},$ we get
$\int\sqrt{(\text{x}-1)^2+(2)^2}\text{dx}$ $=\frac{\text{x}-1}{2}\sqrt{2^2+(\text{x}+1)^2}+2\log\Big|\text{x}-1+\sqrt{2^2+(\text{x}-1)^2}\Big|+\text{C}$
$\Rightarrow\ \int\sqrt{5-2\text{x}+\text{x}^2}\text{dx}$ $=\frac{\text{x}-1}{2}\sqrt{5-2\text{x}+\text{x}^2}+2\log\Big|\text{x}-1+\sqrt{5-2\text{x}+\text{x}^2}\Big|+\text{C}$

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