Q.State the Fundamental Theorem of Integral Calculus, and use it to evaluate $\displaystyle\int_{1}^{2}(3x^{2}+2x)\,dx$.

The Fundamental Theorem says a definite integral equals $F(b)-F(a)$ for any antiderivative $F$; find one antiderivative here and evaluate it at both limits.

Statement of the theorem (no proof required at this level): if $f(x)$ is continuous on $[a,b]$ and $F(x)$ is any antiderivative of $f(x)$, then $\displaystyle\int_{a}^{b}f(x)\,dx=F(b)-F(a)$.

Applying it here: find an antiderivative of $f(x)=3x^{2}+2x$ using standard formula 1 on each term:

(the constant $C$ is not needed for a definite integral, since it would cancel — see this chapter's own note on why). …

Adding a $+C$ inside the definite-integral evaluation, which is unnecessary since it always cancels in $F(b)-F(a)$. Also, subtracting the limits in the wrong order ($F( …