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Often while solving an equation of the form $$ f(y) \frac{dy}{dx} = g(x) $$ we integrate on both sides to obtain $$ \int f(y) \frac{dy}{dx} \, dx = \int g(x) \, dx. $$ We normally write it as $$ \int f(y) \, dy = \int g(x) \, dx. $$ But I don't understand how we replace $ \int f(y) \frac{dy}{dx} \, dx $ with $ \int f(y) \, dy $ on the LHS.

Is there a way to show that $$ \int f(y) \frac{dy}{dx} \, dx = \int f(y) \, dy $$

Lone Learner
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    Hint: what are these integrals' $x$-derivatives? – J.G. Nov 28 '21 at 17:42
  • https://en.wikipedia.org/wiki/Integration_by_substitution – PC1 Nov 28 '21 at 17:42
  • There are many similar questions already: https://math.stackexchange.com/questions/27425/what-am-i-doing-when-i-separate-the-variables-of-a-differential-equation, https://math.stackexchange.com/questions/737928/the-formalism-behind-integration-by-substitution, https://math.stackexchange.com/questions/402303/understanding-the-differential-dx-when-doing-u-substitution, etc. – Hans Lundmark Nov 28 '21 at 20:40

1 Answers1

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This is just integration by substitution. Note for $$ \int f(y)y' dx $$ we can set ${u = y(x)}$ to obtain $$ \rightarrow \int f(u)du $$ but ${u=y}$, so $$ \rightarrow \int f(y)dy $$

Riemann'sPointyNose
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