How do you find the integral of x/ (x^2+y^2)?

 To find the integral of \( \frac{x}{x^2 + y^2} \) with respect to \( x \), you can treat \( y \) as a constant and use the substitution method. Let's denote \( u = x^2 + y^2 \). Then, \( du = 2x \, dx \), and solving for \( dx \), we get \( dx = \frac{du}{2x} \).

Substituting \( u \) and \( dx \) into the integral, we get:

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

Now, integrating \( \frac{1}{u} \) with respect to \( u \), we get:

\[ \frac{1}{2} \ln|u| + C \]

Finally, substituting back \( u = x^2 + y^2 \), we get:

\[ \frac{1}{2} \ln|x^2 + y^2| + C \]

So, the integral of \( \frac{x}{x^2 + y^2} \) with respect to \( x \) is \( \frac{1}{2} \ln|x^2 + y^2| + C \), where \( C \) is the constant of integration.