Anti-derivatives...?

Question

Consider the function f(t)=8sec^2(t)–10t^2. Let F(t) be the antiderivative of f(t) with F(0)=0.
Then F(3)= ???

Thanks for any help/suggestions/hints/solutions (along with explainations of how you reached that solution)...
I appreciate it!

Answer 1

F(t)=8tan(t)-(10/3)t^3+C

F(0)=C=0

So F(t)=8tan(t)-(10/3)t^3

F(3)=8tan(3)-90

Answer 2

An antiderivitive is the same as the integral of a function. So you have to integrate f(t), which is not terribly hard; it is done term by term. You get an extra undetermined constant, which you evaluate by setting F(0)=0 to solve. Then you can substitute 3 for x in the final function and come up with the answer.

Answer 3

Just do a direct integration!

int f(t) = 8 int sec^2(t) - 10 int t^2
= 8 tan(t) - 10/3 t^3 + C (check int sec^2, I think its tan)

F(t) = 8 tan(t) - 10/3 t^2 + C. says F(0) = 0, or

F(0) = 8 tan(0) - 10/3(0^2) + C = 0, so C = 0, so

F(t) = 8 tan(t) - 10/3 t^2.

F(3) = 3 tan(3) - 10/3 * 9
= 3 tan(3) - 30.

Done.