Question Detail

Let's analyze each option:A. $\int f(x)dx = 0$: This statement is generally not correct. The indefinite integral $\int f(x)dx$ represents the family of all antiderivatives of $f(x)$, which is $F(x) + C$. This is a function, not necessarily 0, and not a single value. For example, $\int 1 dx = x + C$, which is not 0. Even if $f(x)=0$, $\int 0 dx = C$, which is 0 only if $C=0$. So, as a general rule, this statement is false.B. $\int f(x)dx = \int f(x)dx$: This is a trivial statement of equality. An indefinite integral is equal to itself. This is correct.C. $\int_a^b f(x)dx \le \int_a^b f(x)dx$ if $a < b$: This is also trivially true, as any quantity is less than or equal to itself ($X \le X$). The condition $a < b$ is standard for definite integrals. This is correct.D. $\int_a^b f(x)dx \le \int_a^b g(x)dx$ if $f(x) \le g(x)$ in $[a,b]$: This is a fundamental property of definite integrals, known as the comparison property of integrals. If one function is always less than or equal to another over an interval, its integral over that interval will also be less than or equal. This is correct.Therefore, the statement that is not correct is option A.