# AP Calculus BC - Antideriative of cos(1-x^2)/(x^2 + root(x))

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if someone could explain how to solve this problem, and why the answer is 7.982. It's a calculator problem.

If $f(2) = 8$ and $f '(x) = \frac{\cos(1-x^2)}{(x^2 + \sqrt{x})}$ , what is $f(7) = ?$

• Does the exam cover any numerical integration techniques? – carmichael561 Apr 29 '16 at 5:22
• Was it a multiple choice question? The numerator will always be between $-1$ and $1$ and oscillates in sign. Past 2 the denominator will predominate and the positive and negative changes will average out somewhat. So the value at $7$ will probably not be much different from the value at $2$. So unless there were other values close to $8$ on a multiple choice question, $7.982$ would be a good answer. – John Wayland Bales Apr 29 '16 at 5:38
• @JohnWaylandBales there were, unfortunately. 8.170 was also an answer. Thank you for the reply, though. – Ozymandias Apr 29 '16 at 6:24

From the fundamental theorem, (since $f'$ is continuous on [2,7]) $$\int_2 ^7 f'(x) dx = f(7) - f(2) = f(7) - 8$$
$\int_2 ^7 f'(x) dx \approx -0.0182$ and thus $f(7) \approx 8 - 0.0182 = 7.982$.