Mathematics: Am I using integration by substitution correctly?

Question

Warning: I am a novice, so the working out below may seen hilariously nonsensical. If I want to find x for this equation: 1/(k - x) = dy/dx Integrate (1/(k - x)) dx = y --- (1) Let's say that 1/(k - x) = u, so x = (uk-1)/u dx/du = 1/(u^2) dx = 1/(u^2) du Substituting into equation (1): Integrate 1/u du = y y = ln(x) + C (C = constant of integration) x = e^(y-C) (e = Euler's number) Is the calculation I did above legitimate or just meaningless claptrap?

Answer 1

The differential equation is .

Integrate on both sides.

Let k - x = u

Differentiate on both sides

- dx = du

dx = -du

    (Using Logarithmic Properties )

Therefore .