Question

If $\frac{d}{d x} \int f(x) d x=\frac{d}{d x} \int g(x) d x$, then
Assertion (A) $\int f(x) d x$ and $\int g(x) d x$ are equivalent to each other.
Reason (R) Two indefinite integrals with the same anti-derivative lead to the same family of curves.

• A. Both $A$ and $R$ are correct; $R$ is the correct explanation of $A$
• B. Both $A$ and $R$ are correct; $R$ is not the correct explanation of $A$
• C. $A$ is correct; $R$ is incorrect
• D. $R$ is correct: $A$ is incorrect

Answer 1

Answer: (C)

Two indefinite integrals with the same derivative lead to lead to the same family of curves and so they are equivalent.
Let $f$ and $g$ be two functions such that
$\frac{d}{d x} \int f(x) d x=\frac{d}{d x} \int g(x) d x$
or $\frac{d}{d x}\left[\int f(x) d x-\int g(x) d x\right]=0$
Hence, $\int f(x) d x-\int g(x) d x=C$, where $C$ is any real number
or $ \int f(x) d x=\int g(x) d x+C$
So, the families of curves $\left\{\int f(x) d x+C_1, C_1 \in R\right\}$
and $\left\{\int g(x) d x+C_2, C_2 \in R\right\}$ are identical
Hence, in this sense, $\int f(x) d x$ and $\int g(x) d x$ are equivalent.