$$ \int(\sqrt{\tan x}+\sqrt{\cot x}) d x= $$
$2 \tan ^{-1}\left(\frac{\tan x-1}{\sqrt{\tan x}}\right)+C$
$\tan ^{-1}\left(\frac{\tan x-2}{2 \sqrt{\tan x}}\right)+C$
$\sqrt{2} \tan ^{-1}\left(\frac{\tan x-1}{\sqrt{2 \tan x}}\right)+C$
$\sqrt{2} \tan ^{-1}\left(\frac{\tan x+1}{\sqrt{2} \tan x}\right)+C$
$\int \frac{\sqrt{x-2}}{2 x+4} d x=$
$\sqrt{x-2}-\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{x-2}}{2}\right)+C$
$\sqrt{x-2}-2 \tan ^{-1}\left(\frac{\sqrt{x-2}}{2}\right)+C$
$\sqrt{x-2}+2 \tan ^{-1}\left(\frac{\sqrt{x-2}}{2}\right)+C$
$\sqrt{x-2}+\frac{1}{2} \tan ^{-1}\left(\frac{\sqrt{x-2}}{2}\right)+C$
If $\int x^{49}\left[\tan ^{-1} x^{50}+\frac{x^{50}}{1+x^{100}}\right] d x=\frac{x^n}{k} f(x)+c$, then
$$ f(x)-f\left(\sqrt[k]{x^n}\right)= $$
$k+n$
$k-n$
$1 / k$
$1 / n$
$$ \int \frac{x}{\sqrt{x^2-2 x+5}} d x= $$
$\sqrt{x^2-2 x+5}+\sinh ^{-1}\left(\frac{x-1}{2}\right)+C$
$\frac{1}{2} \sqrt{x^2-2 x+5}+\sin ^{-1}\left(\frac{x-1}{2}\right)+C$
$2 \sqrt{x^2-2 x+5}+\cosh ^{-1}\left(\frac{x-1}{2}\right)+C$
$\sqrt{x^2-2 x+5}-\cos ^{-1}\left(\frac{x-1}{2}\right)+C$
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