Integrate the function 1 (2-x)^ 2 +1 - Vidyadip

Question

Integrate the function $\frac{1}{\sqrt{(2-x)^{2}+1}}$

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Answer

Let 2 - x = t
$\Rightarrow$ -dx = dt
$\Rightarrow \int \frac{1}{\sqrt{(2-x)^{2}+1}} d x=-\int \frac{d t}{\sqrt{t^{2}+1}}$
$=-[\log |\mathrm{t}+\sqrt{\mathrm{t}^{2}+1}|]+\mathrm{C}$
$\begin{equation} =\log |t+\sqrt{t^{2}+1}|^{-1}+C \end{equation}$
$\begin{equation} =\log \frac{1}{|t+\sqrt{t^{2}+1}|}+C \end{equation}$
$=\log \left|\frac{1}{(2-x)+\sqrt{x^{2}-4 x+5}}\right|+C$

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