Question

Let F(x)=int_{0}^{x}(e)^{{(3t})^{4}}

Find the Maclaurin polynomial polynomial of degree 5 for F(x)

Answer 1

\[Let F(x)=\int\limits_{0}^{x}(e)^{{(3t})^{4}} \]

Answer 2

that is what it looks like

Answer 3

Integral of e^((3t)^4) dx.

Start with Maclaurin's series for e^x:

e^x = 1+x +x^2/2! + x^3/3! + x^4/4! + x^5/5! +....

for e^((3t)^4 substitute (3t)^4 for x:

F(t) = 1 + (3t)^4 + ((3t)^4)^2/2! + ((3t)^4)^3/3! + ((3t)^4)^4/4! ...

= 1 +81t^4 + (6561t^8)/2 +(177147t^12)/2 +
(14348907t^16)/8 + (1162261467t^20)/40 + (31381059609t^24)/80 + ...

You just need to integrate term by term as required.