Question

F is a differentiable function for all x. Which of the following statements must be true?

a. d/dx ∫ f(x)dx=f(x)
b. d/dx ∫ f(t)dt= - f(x)
c. ∫ f′(x)dx=−f(x)

Answer 1

The limits in the integral of the options are missing. They are :

a).
`(d)/(dx) \int^2_0 f(x)dx = 0`

b).
`(d)/(dx) \int^x_2 f(t)dt = 0`

c).
`$\int^x_2f'(x)dx=f(x)$`

Solution:

We known that

`$(d)/(dx)\int^(g(x))_(h(x))f(t)dt = f(g(x))\cdot g'(x)-f(h(x))\cdot h'(x)$`

a).
`(d)/(dx) \int^2_0 f(x)dx`

`$=f(2) \cdot (d)/(dx)(2) - f(0)\cdot (d)/(dx)(0)$`

= 0 - 0

= 0

Hence it is true.

b).
`(d)/(dx) \int^x_2 f(t)dt`

`$=f(x)(d)/(dx)(x)-f(2)\cdot (d)/(dx)(2)$`

`$=f(x) \cdot 1 -f(2)* 0 $`

= f(x)

Hence it is true.

c).
`$\int^x_2f'(x)dx`

`$\left[f(x)\right]^x_2 = f(x)-f(2)$`

`$\\eq f(x) $`

Hence it is false.

Therefore option a) and b). are true.