例子 $\int \sqrt {\frac {1}{1+2x^2}} dx$

这里就记作一个结论吧:

$\begin{aligned} \int \frac{d x}{\sqrt{1+2 x^{2}}} &=\int \frac{d x}{\sqrt{1+(\sqrt{2 x})^{2}}}=\int \frac{\sec (x)}{\sqrt{2}} d x \\ &=\frac{1}{\sqrt{2}} \ln |\tan x+\sec x| \end{aligned}$

其中 $u=\arctan (\sqrt {2} x)$

反代回去,得到

$I =\frac{1}{\sqrt{2}}\ln \left|\sqrt{2}x+\sqrt{1+2x^2}\right|+C$