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# Show that $f(x)=2x+\cot^{-1}x+log\bigg(\sqrt {(1+x^2)}-x\bigg)$ is increasing in R.

Toolbox:
• Let $f(x)$ be a function defined on $(a,b)$. If $f'(x) >0$ for all $x \in (a,b)$ except for a finite number of points, where $f'(x)>0$, then $f(x)$ is increasing on $(a,b)$
Step 1
$f(x)=2x+ \cot^{-1}x+ \log ( \sqrt{1+x^2}-x)$
differentiating w.r.t $X$ we get,
$f'(x)=2-\large\frac{1}{1+x^2}+\large\frac{1}{\sqrt{1+x^2-x}} \bigg( \large\frac{1}{2\sqrt{1+x^2}}.2x-1 \bigg)$
$= 2-\large\frac{1}{1+x^2}+ \bigg(\large\frac{2x}{2\sqrt{1+x^2}}-1 \bigg) \large\frac{1}{\sqrt{1+x^2}-x}$
On simplifying we get
$\large\frac{2x^2+x+2}{2x^2-x+2}- \large\frac{1}{\sqrt{1+x^2}-x} >0$
It is clear that for all values of $x$ the above equation gives only a positive value. Hence the function is increasing in $R$