Antiderivatives

Question

Consider the function 𝑓(𝑡)=8sec2(𝑡)-3𝑡3. Let 𝐹(𝑡) be the antiderivative of 𝑓(𝑡) with 𝐹(0)=0.

Then 𝐹(3)=   

Answer 1

F(t) = ∫ [8sec²t - 3t³]dt, where F(0) = 0

F(t) = 8tant - (3/4)t⁴ + C

Since F(0) = 0, 8tan0 - (3/4)(0)⁴ + C = 0

8(0) - 0 + C = 0. So, C = 0.

F(t) = 8tant - (3/4)t⁴

Plug in t = 3 to get F(3).