Answer:
[tex]\int\limits^0_c f'(x) dx =f(c)-f(0)[/tex]
Step-by-step explanation:
we are given
f has a continuous derivative on [0,c]
and we know that
derivative and integral are inverse of each other
so, we can write as
[tex]\int \:f'\left(x\right)dx=f\left(x\right)[/tex]
we are given bound
so, we can use fundamental theorem of calculus
[tex]\int\limits^a_b f'(x) dx =f(b)-f(a)[/tex]
we have bound from 0 to c
so, we get
[tex]\int\limits^0_c f'(x) dx =f(c)-f(0)[/tex]
By definition of bounded integral, we will get that the integral of f'(x)dx in the given interval is just equal to f(c) - f(0)
Remember that, by definition, f(x) is the antiderivative of f'(x), this means that:
[tex]\int\limits {f'(x)} \, dx = f(x)[/tex]
If f(x) and f'(x) are continuous
Now if we add bounds (remember, we need both to be continuous in the given interval) will just get:
[tex]\int\limits^c_0 {f'(x)} \, dx = f(c) - f(0)[/tex]
Then the integral f'(x)dx in the given interval is just f(c) - f(0)
If you want to learn more, you can read:
https://brainly.com/question/16981829
