Given how complicated the derivative,
[tex]f'(x)=\dfrac{\cos(1-x^2)}{x^2+\sqrt x}[/tex]
looks, integrating it to find [tex]f(x)[/tex] seems futile, and actually not what needs to be done. You're asked to approximate [tex]f(7)[/tex] using a given point and the value of the derivative at any point. To do that, you can use a linear approximation.
[tex]f(7)\approx f(2)+f'(2)(7-2)[/tex]
which comes from exploiting the mean value theorem (solve for [tex]f'(2)[/tex] above to see how)
Then
[tex]f(7)\approx8+\dfrac{5\cos(1-2^2)}{2^2+\sqrt2}\approx7.086[/tex]
The closest answer would be B. The reason the listed answer is so far off is because 7 is not so close to 2.
