Question

Integrate the following with respect to x.

If f'(x) = 8x³ – 2x and f(2) = 8, then find f(x)

Answer 1

f'(x) = 8x³ – 2x

Integrating both sides of the equation,

`int "f'"(x)  "d"x = int 8x^3  "d"x - int 2x  "d"x + "c"`

f(x) = `(8x^4)/4 - (2x^2)/2 + "c"`

f(x) = 2x⁴ – x² + c

Using f(2) = 8,

8 = 2(2)⁴ – (2)² + c

8 = 32 – 4 + c

c = – 20

∴ f(x) = 2x⁴ – x² – 20