Consider the function f(t)=7 sec^2(t) − 4t^2. Let F(t) be the antiderivative of f(t) with F(0)=0. Then F(t) equals ?

Question

Consider the function f(t)=7 sec^2(t) − 4t^2. Let F(t) be the antiderivative of f(t) with F(0)=0.

Then F(t) equals ?

Answer 1

The problem states that F(t) is the antiderivative of f(t); therefore, what one is being asked to find is the integral of the function f(t) with respect to t.

1. F(t) = ∫ f(t) dt
2. F(t) = ∫ ( 7 sec²(t) - 4t² ) dt
3. F(t) = ∫ 7 sec²(t) dt - ∫4t² dt
4. F(t) = 7 ∫ sec²(t) dt - 4 ∫ t² dt
5. F(t) = 7 tan(t) - (4/3)t³ + C

The constant of integration, C, must be found to obtain the final function. Therefore, the values given as the beginning of the problem must be used to evaluate this function. The problem states F(0) = 0, which means that, at t = 0, F(t) = 0. Using these values in the resulting function above and solving for C:

7 tan(t) - (4/3)t³ + C = F(t)

7 (0) - (4/3)(0)³ + C = 0

0 - 0 + C = 0

C = 0

Using this last value of C in the last equation:

F(t) = 7 tan (t) - (4/3)t³ + C

F(t) = 7 tan (t) - (4/3)t³ + 0

The final answer is:

F(t) = 7 tan (t) - (4/3)t³

Answer 2

F(t) = 7tant - 4t³/3 + C; F(0) = C = 0; F(t) = 7tant - 4t³/3