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If f'(x) = \sin\left(\frac{\pi}{2}e^x\right) and $f(0) = 1$ then $f(2) =?$

I'm currently studying for my calculus exam and came across this multiple choice question. I have tried to do $u$ substitution to get $f(x) + c$. And then to just plug in the values. I have been dwelling on this for half an hour now either I'm doing it right and getting the answer wrong I have the wrong approach. Would someone please be kind enough to show a step by step guide for questions like these in general ( so its useful for other people as well).

The choices are:

  • A $-1.819$
  • B $-0.843$
  • C $-0.819 $
  • D $0.157$
  • E $1.157 $
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    @GEdgar: Oops, "*" disappeared somewhere while I copy&pasted the URL... [This is a corrected version](https://www.google.com/search?q=sin(%28pi%2F2%29e%5Ex)+from+-0.5+to+2.5)2012-02-25

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Here is a picture.

enter image description here

The function $f'$ is in red. Consider another function $g$ with $g \le f'$, seen in green. The points on the $x$-axis are 0.6, 1.4, 1.7. The negative parts are $-1$, the positive parts are piecewise linear, with peaks where the curve is maximum, namely at $x=0$ and at $x=\log 5$. If we compute $1+\int_0^2 g(x)dx$ it is easy, two rectangles of known size, two triangles of height 1 and known widths, plus the 1. The answer is 0.35. But this is smaller than the value $1+\int_0^2 f'(x)dx$. The only choice bigger than 0.35 is, therefore, the answer.