Question

Which of the following is true for indefinite integral?

• A. An indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f.
• B. This can be stated symbolically as F = f.
• C. Without upper and lower limits, also called an antiderivative
• D. None of the above

Answer 1

Answer: (A), (C)

The correct options are
A An indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f.
C Without upper and lower limits, also called an antiderivative

To prove that;

$\frac{dF}{dx}=f$

Given,

$\int fdx=F$

By the first principle of differentiation and integration;

$\frac{dF}{dx}=\frac{F(x+dx)-F\left(x\right)}{dx}{\int }_a^{b}fdx=[f\left(a\right)+f(a+dx)+f(a+2dx).....+f(b-2dx)+f(b-dx\left)\right]dx=F\left(b\right)-F\left(a\right)$

Substitute,

$b=x+dxF(x+dx)-F\left(a\right)=[f\left(a\right)+f(a+dx)+f(a+2dx).....+f(x-dx)+f(x\left)\right]dxb=xF\left(x\right)-F\left(a\right)=[f\left(a\right)+f(a+dx)+f(a+2dx).....+f(x-dx)]dx$

$F(x+dx)-F\left(x\right)=\left[F\right(x+dx)-F(a\left)\right]-\left[F\right(x)-F(a\left)\right]=f\left(x\right)dx\therefore \frac{F(x+dx)-F\left(x\right)}{dx}=f\left(x\right)Hence,\frac{dF}{dx}=f\left(x\right)$

So, A is true.

B is false

Indefinite integral is an integral without upper and lower limits and it referred as antiderivative.

Thus, C is true.