Two indefinite integrals with the same derivative

Question

Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent

quoting from my calculus book. ie. $$ \frac{d}{dx}\int f(x)dx=\frac{d}{dx}\int g(x)dx\implies\int f(x)dx\text{ and }\int g(x)dx\text{ are equivalent.} $$ This makes sense as if the derivative of two functions is equal then the functions are separated by a constant. But, using the property $\frac{d}{dx}\int f(x)dx=f(x)$, it seems to conclude that $f(x)=g(x)$

What am I missing here ?

Answer 1

You are not missing anything.

$$\int f(x)dx$$ simply means an antiderivative of $f(x).$

$$\frac{d}{dx}\int f(x)dx=\frac{d}{dx}\int g(x)dx\implies\int f(x)dx =\int g(x)dx +C$$

where C is a constant.

Also,

$$\frac {d}{dx}\int f(x)dx=\frac{d}{dx}\int g(x)dx \implies f(x)=g(x)$$

Answer 2

$\int f(x)dx$ and $\int g(x)dx$ are "families." When $\frac{d}{dx}\int f(x)=\frac{d}{dx}\int g(x)$ then we can say that functions in the family $\int f(x)dx$ are the same as the functions in the family $\int g(x)dx$. This doesn't mean that $f(x)=g(x)$. Take the case $g(x)=f(x)+1$ for example.