TITLE: Finding the value of $b$ so $\int_1^b (x-2)^3 dx =0$
QUESTION [5 upvotes]: Please how do find $b\gt 1$ so that $$ \int_1^b (x-2)^3~dx =0?$$  This question is on a chapter dealing with antiderivatives and I'm not sure how to go about it. At this point it is assumed that I don't know how to integrate yet. I'm also not allowed to use the fundamental theorem of calculus.

REPLY [9 votes]: Note that the substitution $u=x-2$ gives the integral $$\int_{-1}^{b-2}u^3\,du.$$ What do you know about integrals of odd functions (in particular what sort will give an integral of $0$)?
The fact that $u\mapsto u^3$ is strictly monotone is enough to show uniqueness of the value thus determined. (Thanks to Fly by Night for pointing out that something additional was required to prove uniqueness.)