What is the integral of y/x dx?

Question

amistre64 · Answer

we can change this a bit:

dy =  (y/x) dx   or dy/y = dx/x and integrate (S) both sides.

(S) (1/y) dy = (S) (1/x) dx
ln(y) = ln(x) + C
e^(ln(y)) = e^(ln(x) + C)
y=e^(ln(x)) * e^C
y=Cx

This is my best guess :)

gina · Answer

yes you right

amistre64 · Answer

yay!!  :)

anonymous · Answer

Well…really, $$\int\limits(y/x)dx$$means to integrate with respect to x, treating everything else as a constant. Thus, $$y \int\limits(1/x)=y \ln \left| x \right|+C.$$Just sayin'.

amistre64 · Answer

I would love to agree with you carly; but, ...

If we undo that we get:

d(f(x))/dx = d(y ln(x))/dx
f'(x) = y*(1/x)*(dx/dx) + (dydx)*(ln(x))
f'(x) = y/x + y' ln(x). ; which is not the same as (y/x)

Right?

anonymous · Answer

What is the derivative of ln x? Now what's that times a constant y, hmm?

amistre64 · Answer

But the "y" is not a constant; it is an implicit function of "x".  If "y" is an independant variable of f(x,y) then yes, we would hold "y" as a constant to find the rate of change of f(x,y) with respect to "x".