Constant of Integration — Question 9

Question 9 - Solution

Let $F$ and $G$ be two antiderivatives of $f$ on the interval $I$. Then $$F'(x)=f(x) \quad\text{and}\quad G'(x)=f(x) \quad\text{for all }x\in I.$$

Assume that there exists a point $x_0\in I$ such that $$F(x_0)=G(x_0).$$

Consider the function $$H(x)=F(x)-G(x).$$

Differentiate $H(x)$: $$H'(x)=F'(x)-G'(x)=f(x)-f(x)=0 \quad\text{for all }x\in I.$$

Thus, $H$ has zero derivative on $I$, so $H$ is constant on $I$. That is, $$H(x)=K \quad\text{for all }x\in I.$$

Evaluate $H$ at $x_0$: $$H(x_0)=F(x_0)-G(x_0)=0.$$

Hence, $K=0$, and therefore $$F(x)-G(x)=0 \quad\text{for all }x\in I.$$

This implies $$F(x)=G(x) \quad\text{for all }x\in I.$$

$$\boxed{F(x)\equiv G(x)}$$