Constant of Integration — Question 7

Question 7 - Solution

Consider the two functions $$F_1(x)=\int_a^x f(t)\,dt, \qquad F_2(x)=\int_b^x f(t)\,dt.$$

Both $F_1$ and $F_2$ are antiderivatives of $f$ on $I$ by the Fundamental Theorem of Calculus.

Evaluate the difference between them: $$F_1(x)-F_2(x) = \int_a^x f(t)\,dt-\int_b^x f(t)\,dt.$$

Using properties of definite integrals, $$\int_a^x f(t)\,dt-\int_b^x f(t)\,dt = \int_a^b f(t)\,dt.$$

The right-hand side is a fixed real number that does not depend on $x$.

Therefore, $$F_1(x)-F_2(x)=\text{constant}.$$

This shows that changing the lower limit of integration alters the antiderivative only by an additive constant.

$$\boxed{\int_a^x f(t)\,dt=\int_b^x f(t)\,dt+\text{constant}}$$