Consider the following statements (A) and (B) | vedclass

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(D) For statement $(A)$: The left side is $\int_a^b \frac{d}{d x}(f(x)) d x = f(b) - f(a)$,which is a constant value. The right side is $\frac{d}{d x}

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Both $(A)$ and $(B)$ are false

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(D) For statement $(A)$: The left side is $\int_a^b \frac{d}{d x}(f(x)) d x = f(b) - f(a)$,which is a constant value. The right side is $\frac{d}{d x} \int_a^b f(x) d x$. Since $\int_a^b f(x) d x$ is a definite integral resulting in a constant,its derivative with respect to $x$ is $0$. Thus,$f(b) - f(a) 
eq 0$ in general,so $(A)$ is false.For statement $(B)$: By the definition of the indefinite integral,$\frac{d}{d x} \int f(x) d x = f(x)$. The constant $C$ is only present in the result of the indefinite integral itself,not in its derivative. Therefore,$\frac{d}{d x} \left( \int f(x) d x ight) = f(x)$,not $f(x) + C$. Thus,$(B)$ is false.Conclusion: Both $(A)$ and $(B)$ are false.

Consider the following statements $(A)$ and $(B)$:
$(A) \int_a^b \frac{d}{d x}(f(x)) d x = \frac{d}{d x} \int_a^b f(x) d x$
$(B) \frac{d}{d x} \left( \int f(x) d x \right) = f(x) + C$
Which one of the following is true?

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Consider the following statements $(A)$ and $(B)$:$(A) \int_a^b \frac{d}{d x}(f(x)) d x = \frac{d}{d x} \int_a^b f(x) d x$$(B) \frac{d}{d x} \left( \int f(x) d x \right) = f(x) + C$Which one of the following is true?