TITLE: How to solve $a$ in $\int_a^xf\left(t\right)dt=x^2-2x-3$
QUESTION [6 upvotes]: How do you solve for $a$ in $\int_a^xf\left(t\right)dt=x^2-2x-3$?
I have differentiated both sides with respect to $x$, but do not know how to continue after this.

REPLY [5 votes]: Method 1 (your approach)
Differentiating both sides with respect to $x$, $f(x) = 2x-2$.
And thus, $\int_a^xf\left(t\right)dt=\int_a^x\left(2t-2\right)dt=x^2-2x-(a^2-2a)$
Comparing this to the original equation, $x^2-2x-3=x^2-2x-(a^2-2a)$
You can solve it from here.

REPLY [2 votes]: Well after differentiating you know:
$$f(x) = 2x-2$$ and therefore $\int_a^x 2t-2 dt = t^2-2t|_a^x = x^2-2x-a^2+2a$.
From this immediately follows: $a^2-2a = 3$ with the solutions $a=-1$ and $a=3$