Q.Evaluate $\displaystyle\int_{0}^{2} (3x^{2}+4)\,dx$ using the Fundamental Theorem of Integral Calculus.

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.

Find an antiderivative of $f(x)=3x^{2}+4$ using the power rule on each term:

(the constant $C$ is not needed for a definite integral, since it cancels).

Evaluate at the upper and lower limits:

$F(2)=2^{3}+4(2)=8+8=16, \qquad F(0)=0^{3}+4(0)=0$ …

Adding a $+C$ inside the definite-integral evaluation, which is unnecessary since it always cancels in $F(b)-F(a)$. Also, forgetting the constant term $4$ when finding t …