Integration after implicit differentiation

[soopo]

Homework Statement

What is the integral of:
$$y + x \frac{dy}{dx}$$

The Attempt at a Solution

I know that it is xy, after implicit differentiation.
However, I cannot get it without prior knowledge of implicit differentiation.

 

[HallsofIvy]

Homework Statement

What is the integral of:
$$y + x \frac{dy}{dx}$$

The Attempt at a Solution

I know that it is xy, after implicit differentiation.
However, I cannot get it without prior knowledge of implicit differentiation.

Note that you would write the integral of that function as
$$\int (y+ x\frac{dy}{dx})dx= \int ydx+ xdy$$
That means you are looking for a function F(x,y) so that $dF= F_x dx+ F_y dy= ydx+ xdy$. If $F_x= y$ then integrating with respect to x, while keeping y constant, F(x,y)= xy+ g(y) where, because we are treating y as a constant, the "constant of integration" may depend on y: g(y). From that, $F_y= x+ g'(y)= x$ which tells us that g'(y)= 0. That means that g really is a constant: g= C so F(x,y)= xy+ C for any constant C.

 

[soopo]

Note that you would write the integral of that function as
$$\int (y+ x\frac{dy}{dx})dx= \int ydx+ xdy$$

Do you mean this?
$$\int (y+ x\frac{dy}{dx})dx= \int ydx+ \int xdy$$

 

[NoMoreExams]

Isn't that what he wrote?

 

[soopo]

Isn't that what he wrote?

I have had an idea that the effect of integral sign ends at the first dx or dy sign. Perhaps, this is not true.

 

[soopo]

Thank you, Hallsofivy!