Question

Find the anti-derivative $F$ of $f$ defined by $f\left(x\right)=4x^3-6,$ where $F\left(0\right)=3$

Answer 1

Given: $f\left(x\right)=4x^3-6$
Given that $F$ is anti derivative $f$
So, $F\left(x\right)=\int f\left(x\right)dx$
$=\int (4x^3-6).dx$
$=\int 4x^{3}dx-\int 6dx$
$=4\left(\frac{x^{3+1}}{3+1}\right)-6x+C$
$=x^4-6x+C$
Where $C$ is constant of integration
So, $F\left(x\right)=x^4-6x+C$
Given $F\left(0\right)=3$
$\Rightarrow 3=0+0+C$
$C=3$
$\therefore F\left(x\right)=x^4-6x+3$