Hello, just solve the integral and apply MacLaurin series. And in order to calculate the define integral just replace the values.
Don't solve the integral (it cannot be done in terms of elementary functions, anyway)! You only need the value of F at x= 0 which is obviously 0 because for any f.
Then use the Fundamental Theorem of Calculus to find F'(x) and differentiate that to get F", F''', F'''', and F'''''.