Let .
Find the MacLaurin polynomial of degree 5 for .
Use this polynomial to estimate the value of .
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 $\displaystyle \int_0^0 f(x)dx= 0$ for any f.
Then use the Fundamental Theorem of Calculus to find F'(x) and differentiate that to get F", F''', F'''', and F'''''.