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v(x(t)), where v represents velocity and is a function of position which is a function of time.

I have the equation: v dv/dx = 20x + 5 and want to solve for velocity. The way our professor solved it was by multiplying both sides by dx and integrating => ∫v dv = ∫20x+5 dx. I know doing this is non-rigorous since dv/dx can't be treated as a fraction. So how would I "correctly" do this?

Generally ∫ ƒ(y) dy/dx dx = ∫ f(u) du by substitution or the "reverse" chain rule since dy/dx is the derivative of the composite function y inside f In my case however, ∫ v dv/dx , dv/dx is not the derivative of the composite inside v. How would I solve for v without doing something like multiplying both sides by dx?

I have the equation: v dv/dx = 20x + 5 and want to solve for velocity. The way our professor solved it was by multiplying both sides by dx and integrating => ∫v dv = ∫20x+5 dx. I know doing this is non-rigorous since dv/dx can't be treated as a fraction. So how would I "correctly" do this?

Generally ∫ ƒ(y) dy/dx dx = ∫ f(u) du by substitution or the "reverse" chain rule since dy/dx is the derivative of the composite function y inside f In my case however, ∫ v dv/dx , dv/dx is not the derivative of the composite inside v. How would I solve for v without doing something like multiplying both sides by dx?

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