Find the Maclaurin Polynomial of degree 7

Question

Let F(x) = ∫ (from 0 to x) sin⁡(7^t(2)) dt.

Find the MacLaurin polynomial of degree 7 for F(x).

Use this polynomial to estimate the value of ∫ (from 0.62 to 0) sin⁡(7x^(2)) dx.

Note: your answer to the last part needs to be correct to 9 decimal places.

Answer 1

sin(7t²)=[7t²]-([7t²]³/3!)+([7t²]⁵/5!)-([7t²]⁷/7!)+/-...

taken to infinity the series does equal function and the radius of convergence for t is not restricted. Where t is a real number

F(x)=(7/3)x³-[(7³/7)x⁷]/3!+[(7⁵/11)x¹¹]/5!-[(7⁷/15)x¹⁵]/7!+/-.....+C

evaluating from a to b eliminates the constant

at x=0, F(x)=0+C

use equation to evaluate value at x=.62

since going in negative direction x=.62 to x=0, use negative of value