Answer:
Step-by-step explanation:
Given [tex]f(x) =\int\limits^x_2 {(t^{3}+4t-2 }) \, dt[/tex], to express f as a function of x, we need to first integrate the function given with respect to t as shown;
[tex]f(x) =\int\limits^x_2 {(t^{3}+4t-2 }) \, dt\\\\f(x) = [\frac{t^{4} }{4} + \frac{4t^{2} }{2}-2t]\left \ {t =x} \atop {t=2}} \right.[/tex]
[tex]f(x) = [\frac{t^{4} }{4} + 2t^{2}-2t]\left \ {t =x} \atop {t=2}} \right.\\when\ t = x\\f(x) = \frac{x^{4} }{4} + 2x^{2}-2x\\\\[/tex]
[tex]when\ t = 2;\\f(2) = \frac{2^{4} }{4} + 8-4\\f(2) = 8\\[/tex]
[tex]f(x) - f(2) = \frac{x^{4} }{4} + 2x^{2}-2x -8\\\\at\ x =2; f(x) = \frac{2^{4} }{4} + 8-4 -8 =0\\\\at\ x =4; f(x) = \frac{4^{4} }{4} + 32-8 -8 =80\\\\at x = 8; f(x) = \frac{8^{4} }{4} + 4(8^{2})-16 -8 =1256[/tex]
The function of x at x = 2, 4 and 8 are 0, 80 and 1256 respectively
