How to find the equation? (Integral problem)

Question

Here is a problem in our homework, I've never seen an integral written like that.. what does this question mean and how to solve it? Help please!! :(

Given that $\displaystyle 3x^2-12=\int_a^xf(t)~dt$, find a formula for $f(x)$ and a value of $a$.

Thanks a lot!

Answer 1

HINT: This is a problem about the fundamental theorem of calculus. It says that if

$$F(x)=\int_a^xf(t)~dt\;,$$

then $F'(x)=f(x)$. You’ve been given $F(x)$; use the fundamental theorem to find the $f(x)$. To find $a$, notice that $\int_a^xf(t)~dt=0$ when $x=a$. For what $x$ is your $F(x)$ equal to $0$?

Answer 2

As @tetori noted, Take the derivative of both sides. This is allowed becuase the conditions of Fundmental Theorem satisfied for it. So $$(3x^2-12)'=\left(\int_a^x f(t)dt\right)'=f(x)$$ So $$6x=f(x)$$ Now we have $$3x^2-12=\int_a^x 6tdt=3t^2\big|_a^x=3x^2-3a^2$$ So $3a^2=12$.