what does F(t) equal???

Question

f(t)=6sec^2(t)-3t^3

Let F(t) be the antiderivative of f(t) with F(0)=0. Then F(t) equals?

Answer 1

To find F(t), one must take the integral of f(t) dt.

Integrating f(t) dt, we get…
6 * tan(t) – 3t^4 / 4 + C

Since the integral of sec^2(t) = tan (t) and the integral of t^3 = t^4 / 4. The constant coefficients in front of each term remain the same.

Now we have a general solution to F(t), but we want to find the specific solution to our case and remove the unknown constant, C, from the equation. We are given an initial condition for F(t), we know that F(0) = 0.
Plugging in 0 in to our general equation for F(t), we get,
6 * tan *(0) – 3 (0)^4 / 4 + C = 0
The tangent of zero is zero, and zero to the fourth power is zero. So 0 + 0 + C = 0, so C must equal zero too.

So our final solution for F(t) is,
F(t) = 6 * tan (t) – 3 * t^4 / 4

Answer 2

Integral f(t) dx