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Johnny5

dt ramblings

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Sorry for a bit of necromancy, but this post has an obvious and useful answer.

 

The answer being that the shorthand you're doing to get your answer is disguising the real maths.

 

What you're actually doing is saying:

 

dy/dx= x/y, therefore

 

ydy/dx=x

 

hence if we integrate both sides with respect to x we retain equality (up to adding a constant). You do not actally split it as a fraction.

 

int y(dy/dx)dx = int xdx

 

 

Also, your definite integrals magically become indefinite, by the way.

now we apply the notion that integration is anti-differentiation, since we know that y^2 differentiates to 2ydy/dx.

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