To solve the integral of the expression \((v_x)(u_x) , dx\), we can represent the functions \(v\) and \(u\) as functions of \(x\). Let's assume that \(v = v(x)\) and \(u = u(x)\). Then, \(v_x\) and \(u_x\) denote the derivatives of \(v\) and \(u\) with respect to \(x\).
The integral can be expressed as:
\[ \int v_x u_x , dx \]
To evaluate this integral, we can use integration by parts or recognize if there is a product rule involved. The product rule states that:
\[ \frac{d}{dx}(v u) = v_x u + u_x v \]
However, our goal is to integrate \(v_x u_x\). One strategy is to represent it as a derivative of a product of two terms. Ultimately, though, the integral depends significantly on the specific forms of \(v\) and \(u\).
If we know specific forms for \(v\) and \(u\), we could proceed with further calculations. Otherwise, the expression itself does not simplify directly without additional information. If the integral is to be left in a generic form, we may need to write it as:
\[ \int v_x u_x , dx = \frac{1}{2} u v_x^2 + C \]
This gives the general approach to solving such integrals, but more context about \(u\) and \(v\) would help provide a more precise answer. Please provide the specific functions or any additional information for a more tailored solution!