Q.Verify that $F(x) = \dfrac{x^{3}}{3}+2x$ is an antiderivative of $f(x)=x^{2}+2$, and hence write down $\displaystyle\int (x^{2}+2)\,dx$.

An antiderivative check simply differentiates the candidate function and compares the result with the given $f(x)$; the indefinite integral then adds the constant $C$.

Differentiate $F(x)$: $F'(x)=x^{2}+2$, which equals $f(x)$ exactly — so $F(x)$ is indeed an antiderivative.

Step 1 — differentiate the candidate $F(x)$:

Step 2 — compare with $f(x)$: $F'(x)=x^{2}+2$ matches $f(x)=x^{2}+2$ exactly, confirming $F(x)$ IS an antiderivative of $f(x)$.

Step 3 — write the indefinite integral: since $F(x)$ is one antiderivative, every antiderivative of $f(x)$ differs from it only by a constant:

Confirming $F'(x)=f(x)$ correctly but then forgetting to append $+C$ when writing the indefinite integral itself — the verification step and the general indefinite-integral statement are two separate things, and only the second one needs $C$.