Suppose $\displaystyle f(0)=0$ and $\displaystyle \int_0^2{f'(2t)e^{f(2t)}}dt=5$. Find the value of f(4).

So I first recognized that $\displaystyle \frac{1}{2}e^{f(2t)}$ should be the antiderivative of the integral by FTC so that $\displaystyle \frac{1}{2}(e^{f(4)}-e^{f(0)})=5$.

Since f(0)=0, the equation becomes: $\displaystyle e^{f(4)}-1=10$

Isolating and then taking the natural log of both sides yields: f(4)=ln11

Is that right?

My other question is this: evaluate $\displaystyle \lim_{h\to0}} \frac{1}{h}\int_x^{x+h}e^{arctan{t}}dt$ It was a bonus question on a quiz and I'm wondering how to do it. The thing sort of looks the definition of the derivative?

The first thing I notice is that the variable in the integrand isn't with respect to x. If I rewrote it through FTC, I'd put $\displaystyle \lim_{h\to0}}\frac{1}{h}{e^{arctan(x+h)}}$? I'm not really sure how to do this question. It sort of looks like it could be in the form of the definition of the derivative?