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\int {\frac  {2x}{x^{2}+1}}\,dx\,

To find this integral, start by making a substitution.

Let u=x^{2}+1\,

Then du=2xdx\,

Now solve for dx\, to get

dx={\frac  {du}{2x}}\,

Now substitute these functions into the integral.

\int {\frac  {2x}{x^{2}+1}}\,dx=\int {\frac  {2x}{u}}{\frac  {du}{2x}}=\int {\frac  {du}{u}}\,

Now, this integral is of a form of which we already know the technique.

\int {\frac  {du}{u}}=\ln |u|+C\,

The final step is to undo the substition. Plug x^{2}+1\, back in wherever there is a u\,. Also, note the absolute value bars can be dropped because x^{2}+1\, is always positive. So our answer is


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