As is well known pretty much all integration rules come from differentiation rule, so the substitution rule comes from the chain rule, and the integration by parts comes from the product rule, indeed $$(uv)'=uv'+vu'\implies\int uv'dx=uv-\int vu'dx.$$ But I never saw a rule that used the Over Rule $$(u/v)'=\frac{u'v-v'u}{v^2}=\frac{u'v}{v^2}-\frac{v'u}{v^2}=\frac{u'}{v}-\frac{uv'}{v^2}\implies \int \frac{u'}{v}dx=\frac uv+\int\frac{uv'}{v^2}dx$$ Is this like a new rule or does it already exist?
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2I guess it's hard to use because it's hard to recognize when $\frac{uv'}{v^2}$ is simpler than $\frac{u'}{v}$ – Gregory Grant Dec 26 '15 at 21:16
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Following @GregoryGrant , we're all familiar with the fact that $\frac a b = \frac{ a \cdot c}{b \cdot c}$. But it's also true that $\frac a b = \frac{a \div c}{b \div c}$, but you've probably never seen it (at least formalized) before because it's so rarely useful. – pjs36 Dec 26 '15 at 21:19
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Your "new" rule is in fact the "old" integration by parts rule, in your identity $$ \int \frac{u'}{v}=\frac uv+\int\frac{uv'}{v^2} $$ you may just put $$ U=u \quad \text{and}\quad V=\frac1v $$ to see it: $$ \int U'V=UV-\int UV' $$ since $V'=-\dfrac{v'}{v^2}$.
Olivier Oloa
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It is equivalent to a partial integration. Take the standard partial integration rule and take $v\to v^{-1}$.
user26977
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The quotient rule is just a combination of the product rule and the chain rule, writing $u/v$ as $u\cdot (1/v)$. Similarly, your rule is just a special case of integration by parts, where you write $u'/v$ as $u'\cdot (1/v)$. Explicitly, if you take the formula $$\int uv'dx=uv-\int vu'dx$$ and substitute $1/f$ for $u$ and $g$ for $v$ and then use the chain rule to evaluate $(1/f)'$, you get $$\int \frac{g'}{f}dx=\frac gf+\int\frac{gf'}{f^2}dx,$$ which is exactly your rule (with different variable names to avoid confusion).
Eric Wofsey
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