Constant of Integration — Question 5

Question 5

Suppose that $f$ and $g$ are continuous on a nondegenerate interval $I$, that $a\in I$, and that $$\int_a^x f(t)\,dt = \int_a^x g(t)\,dt \quad\text{for all }x\in I.$$ Prove that $$f(x)=g(x) \quad\text{for all }x\in I.$$

Question 5 - Solution

Set $F(x)=\int_a^x f(t)\,dt$ and $G(x)=\int_a^x g(t)\,dt$.

The hypothesis gives $F=G$ on $I$. At every interior point $x$ of $I$, continuity of both integrands allows the Fundamental Theorem of Calculus:

$$f(x)=F'(x)=G'(x)=g(x).$$

At an included endpoint, take limits from the interior and use continuity of $f$ and $g$. Thus

$$\boxed{f(x)=g(x)\quad\text{for every }x\in I.}$$