Question

If f' (x) = 8x³ − 2x, f(2) = 8, find f(x)

Answer 1

$$\begin{gathered} f\text{'}\left(x\right)=8x^3-2xf\left(2\right)=8 \\ f\text{'}\left(x\right)=8x^3-2x \\ \int f\text{'}\left(x\right)dx=\int \left(8x^3-2x\right)dx \\ =8\int x^{3}dx-2\int xdx \\ f\left(x\right)=8\left[\frac{x^4}{4}\right]-2\times \frac{x^2}{2}+\mathrm{C} \\ f\left(x\right)=2x^4-x^2+\mathrm{C} \\ f\left(2\right)=8\left(\mathrm{Given}\right) \\ f\left(2\right)=2\times 2^4-2^2+\mathrm{C} \\ 8=32-4+\mathrm{C} \\ \mathrm{C}=-20 \\ \therefore f\left(x\right)=2x^4-x^2-20 \end{gathered}$$