alleryIf $F$ is the antiderivative of $f(x)=4x^3-6$ with $F(0)=3$, then $F(x)$ is
A$x^4-6x$
B$4x^4-6x+3$
C$x^4-6x-3$
D$x^4-6x+3$
Answer & Solution
Correct answer: D. $x^4-6x+3$
1. $\int (4x^3-6)\,dx=x^4-6x+C$.
2. Apply $F(0)=3$: $0-0+C=3$, so $C=3$.
3. Hence $F(x)=x^4-6x+3$.
4. Option A ignores the initial condition; check $\dfrac{d}{dx}(x^4-6x+3)=4x^3-6$.
_Source: syllabus Mathematics Ch 7 "Integrals", p.295_
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