( int \frac { 2 x } { x ^ { 2 } + 3 } d x )

Explanation:

Step1: Let ( u = x^{2}+3 )

Differentiate ( u ) with respect to ( x ): ( du=2x dx )

Step2: Substitute into the integral

The integral ( \int\frac{2x}{x^{2}+3}dx ) becomes ( \int\frac{du}{u} )

Step3: Integrate ( \int\frac{du}{u} )

Using the formula ( \int\frac{1}{u}du=\ln|u| + C ), we get ( \ln|u|+C )

Step4: Substitute back ( u = x^{2}+3 )

We have ( \ln\left(x^{2}+3\right)+C ) (since ( x^{2}+3>0 ) for all real ( x ), the absolute - value can be removed)

Answer:

(\ln\left(x^{2}+3\right)+C)