Evaluate exactly, using the Fundamental Theorem of Calculus: ∫[b 0]((x^2)/(3)+4x)dx=?

Answer 1

let f(x) = x²/3 + 4x, then by the Fundamental Theorem of Calculus where F(x) = ∫f(x) then F'(x) = f(x) so we are looking for the function F whose derivative is x²/3 + 4x.

By the power rule, the function F (whose derivative is x²/3 + 4x) would have to be an x³ and an x² function since the power rule reduces the exponent by 1. But notice that if it was just x³ and x² then the derivative of that would be 3x² and 2x while f(x) is x²/3 + 4x. That means F(x) must be composed of x³/9 and 2x² so the derivative turns out right.

The Fundamental Theorem of Calculus also states that ₀∫^b f(x)dx = F(b) - F(0) therefore we can say ₀∫^b f(x)dx = (b³/9 + 2b²) - (0³/9 + 2(0)²) which is just b³/9 + 2b²