Evaluate the following integral: 1 (2-x)^2+1 dx - Vidyadip

Question

Evaluate the following integral:
$\int\frac{1}{\sqrt{(2-\text{x})^2+1}}\text{ dx}$

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Answer

Let $\text{I}=\int\frac{1}{\sqrt{(2-\text{x})^2+1}}\text{ dx}$
Let $2-\text{x}=\text{t}$
$-\text{dx}=\text{dt}$
$\text{dx}=-\text{dt}$
So, $\text{I}=-\int\frac{1}{\sqrt{\text{t}^2+(1)^2}}\text{ dt}$
$\text{I}=-\log\big|\text{t}+\sqrt{\text{t}^2+1}\big|+\text{C}$ $\Big[$Since $\int\frac{1}{\sqrt{\text{x}^2+\text{a}^2}}\text{ dx}=\log\big|\text{x}+\sqrt{\text{x}^2+\text{a}^2}\big|+\text{C}\Big]$
$\text{I}=-\log\big|(2-\text{x})+\sqrt{(2-\text{x})^2+1}\big|+\text{C}$ $[$Since $\text{t}=(2-\text{x})]$

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