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

Question

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

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Answer

$\int\frac{1}{\sqrt{(2-\text{x})^2}+1}\text{ dx}$
$=\frac{\log\bigg|(2-\text{x})+\sqrt{(2-\text{x})^2+1^2}\bigg|}{-1\rightarrow\text{Coeff. of x}}+\text{c}$$ \ \ \ \ \ \ \ \bigg[\because\int\frac{1}{\sqrt{\text{x}^2+\text{a} ^2}}\text{ dx}=\log\bigg|\text{x}+\sqrt{\text{x}^2+\text{a}^2}\bigg|\bigg]$
$=-\log\bigg|2-\text{x}+\sqrt{4+\text{x}^2-4\text{x}+1}\bigg|+\text{c}$
$=\log\Bigg|\frac{1}{2-\text{x}\sqrt{\text{x}^2-4\text{x}+5}}\Bigg|+\text{C}$

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