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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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}

Vedclass · Accepted Answer

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

Vedclass · Answer

(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?

Similar Questions

If $a > 0, b > 0$ and $\int \frac{1}{ax^2+b} dx = \frac{1}{\sqrt{6}} \tan^{-1} \left(\frac{\sqrt{2}x}{\sqrt{3}}\right) + c$, then $\int \frac{1}{bx^2+a} dx = \dots$

Evaluate the integral $\int {\left( {\sin x \cos x \cos 2x \cos 4x \cos 8x} \right) dx}$.

Find the integral of the function $\frac{\cos x}{1+\cos x}$.

Find the following integral: $\int \frac{d x}{3 x^{2}+13 x-10}$

Find the integral of the function $\frac{\sin ^{3} x+\cos ^{3} x}{\sin ^{2} x \cos ^{2} x}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module

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?