Final answer:
To evaluate the integral of (v/-(g+kv)) with respect to v, where g and k are constants, a u-substitution is applied, resulting in the integral being equal to -ln|g+kv|/k plus a constant of integration.
Step-by-step explanation:
The question is asking to evaluate the integral of (v/-(g+kv)) with respect to v, where g and k are constants. This is a routine calculus problem involving integration. Firstly, we will rewrite the integral by factoring out the negative sign, which gives us -∫(v/(g+kv)) dv. Now, to integrate this function, let's perform a u-substitution where u = g + kv and du = k dv. The integral thus becomes -1/k ∫(1/u) du, which is equal to -ln|g+kv|/k plus a constant of integration C.
This problem utilizes the concept of indefinite integration, specifically with algebraic manipulation and u-substitution. It is an example demonstrating the fundamental technique of integration in calculus, which is very common in physical sciences and engineering to calculate quantities like work, potential energy, and other varying quantities.