Calculus 2 MacLaurin Series

Question

Let F(x) = ∫ (from 0 to x) e^{-3t^4} dt

Find the MacLaurin polynomial of degree 5 for F(x).

Answer 1

We can expand f(t) = e ^{- 3t^4} _{= 1 +} (- 3t⁴)/1! + (- 3t⁴)²/2! + ... + (-3t⁴)ⁿ/n! + ...

So e ^{- 3t^4} = 1 -3t⁴ + 9/2t⁸ -... + (- 3)ⁿt⁴ⁿ/n! + ...

Now we integrate to get F(x) = x -3x⁵/5 + 1/2x⁹ - ... + (- 3)ⁿx⁴ⁿ⁺¹/[(4n + 1)n!] + ...