Answer
As shown by the two fundamental theorems of calculus, derivative and integral are inverse processes. See solution below for detailed answer.
Work Step by Step
For a function $f$ continuous on the interval $[a,b]$, the following are true:
1. If $g(x)=\int_a^xf(t)dt$, then $g'(x)=f(x)$
2. $\int_a^bf(x)dx=F(b)-F(a)$, where $F'=f$ ($F$ is the antiderivative of $f$)
From the FTOC (Fundamental Theorem of Calculus) 1,
$\frac{d}{dx}\int_a^xf(t)dt=f(x)$. Therefore, by differentiating the integral of a function, we arrive at the original function.
From the FTOC 2,
$\int_a^bf(x)dx=F(b)-F(a)$. Therefore, by integrating the differentiation of a function, we get the original function in the for $F(b)-F(a)$.
Thus, these two fundamental theorems of calculus show that the derivative and integral are inverse processes.
