Constant of Integration — Question 8

Question 8 - Solution

Since $$\int f(x)\,dx = F(x)+C,$$ the function $F(x)$ is an antiderivative of $f$.

Given that $$F'(x)=0 \quad\text{for all }x\in I,$$ it follows from the Mean Value Theorem that $F$ is constant on the interval $I$.

Thus, there exists a real number $K$ such that $$F(x)=K \quad\text{for all }x\in I.$$

Substitute this into the expression for the indefinite integral: $$\int f(x)\,dx = K + C.$$

Since $K+C$ is itself a constant, we may write $$\int f(x)\,dx = C'.$$

Therefore, if the derivative of an antiderivative is zero on an interval, the indefinite integral reduces to a constant.

$$\boxed{\int f(x)\,dx = C}$$