Simpson's Error Bound Estimation

Question

The problem:

I need to use Error Bound to find n (least) to the $10^{-9}$ in approximating the integral of 5e^x^2 from 0 to 1

I'm using

$$Error(Sn) \le \frac{k(b-a)^5}{180N^4}$$

I found the 4th Derivative to be:

$$ Error^{(4)}(Sn)= 20e^{x^2}(4x^4 + 12x^2 + 13)$$

The max I found to be at 1, so

$$\frac{8060e}{180n^4} \le 0.000,000,001$$

solve for n

$$\left(\frac{8060e}{180\cdot0.000,000,001}\right)^{\frac 14} \le n$$

$$n = 590$$

Is this true? I don't think it is the correct answer, but I'm not sure where my error is

Answer 1

I made a math mistake.

The f''''(1) = 360e (not 8060e)