Is this integral formula valid

Question

So I was playing with the Quotient rule for derivatives and I found $$\int \frac{du}{v}=\frac{u}{v}+\int \frac{u \cdot dv}{v^2}$$ Is this formula valid and does it help to integrate fractions?

Answer 1

This is indeed a valid formula based on the quotient, and has a similar role to integration by parts (which corresponds to the product rule). In fact, just as the quotient rule can be deduced quickly from the product and the chain rule, we can deduce this formula from integrating by parts:

$$\int \frac 1 v \, du = \frac 1 v \cdot u - \int u d \frac 1 v = \frac u v - \int -u \frac{dv}{v^2}$$

as desired.

So it's pretty much exactly as useful as integration by parts is.

Answer 2

Yes, the integral is valid.

$$\int \frac{du}{v}=\frac{u}{v}+\int \frac{u \cdot dv}{v^2}$$

This is a special case of integration by parts which is $$\int udv=uv-\int vdu $$

Integration by parts is a valid and very useful method of interating a product.

What you have done is coming up with a formula which integrates the quotients.

Just as the integration by parts which is not always the key to solve integrals, your formula may or may not be applicable for all quotients.