Suppose G and H are two area functions of a given function f, how to solve for C : G(x)=F(x)+C?

Question

My question : two indefinite integrals of a function being given , how to express one indefinite integral in terms of the other?

Answer 1

$$F(x)=\int_2^xt^2dt\\G(x)=\int_5^xt^2dt\\G(x)=F(x)+C$$ Then just write $C$ from the last equation as $$C=G(x)-F(x)=\int_5^xt^2dt-\int_2^xt^2dt=\int_5^xt^2dt+\int_x^2t^2dt=\int_5^2t^2dt=-\int_2^5t^2dt=-39$$

Answer 2

Note that since $$\int_a^b f(t)\;dt = \int_a^x f(t)\;dt + \int_x^b f(t)\;dt$$ $$ = \int_a^x f(t)\;dt - \int_b^x f(t)\;dt$$ you therefore have $$\int_b^x f(t)\;dt = \int_a^x f(t)\;dt -\int_a^b f(t)\;dt$$