Answer to Question #147641 in Calculus for liam donohue

Question

Consider the function f(x)=5x2−4x7. Let F(x) be the antiderivative of f(x) with F(1)=0.

Then F(x)=

Answer 1

Solution

The anti-derivative or primitive function is attained by integrating the function. Mathematically, it can be denoted as follows;

∫xⁿ= (xⁿ⁺¹)/ (n+1) (+C)

Therefore, we proceed to solve the anti derivative, F(x) of the function f(x) = 5x²−4x⁷

F(x) = 5(x²⁺¹/2+1) – 4(x⁷⁺¹/7+1) + C

F(x) = (5/3) x³ – (4/8) x⁸ + C

F(x) = (5/3) x³ – (½) x⁸ + C

We know that F(1) = 0, we replace F(x) with F(1) in the above function of F(x) to get;

 F(1) = (5/3) 1³ – (½) 1⁸ + C = 0

= 5/3 + ½ + C = 0

Hence, C= –13/6

Therefore, our F(x) becomes;

F(x) = (5/3) x³ – (½) x⁸ –13/6