Q.Verify that $F(x) = \dfrac{x^{4}}{4} - 3x$ is an antiderivative of $f(x)=x^{3}-3$, and hence write down $\displaystyle\int (x^{3}-3)\,dx$.

Checking whether a given function is an antiderivative simply means differentiating it and comparing the result with $f(x)$; the indefinite integral then adds the constant $C$.

Differentiating $F(x)$ gives $F'(x)=x^{3}-3$, which matches $f(x)$ exactly, confirming $F(x)$ is an antiderivative.

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

Step 2 — compare with $f(x)$: $F'(x)=x^{3}-3$ matches $f(x)=x^{3}-3$ 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 general indefinite integral — the verification step and the indefinite-integral statement are two separate things, and only the second one needs $C$.