Finding the MacLaurin polynomial of degree $7$?

Question

The question reads:

let $F(x)=\int_0^x \sin(6t^2) \, dt$

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

2) Use this polynomial to estimate the value of $\int_0^{0.75} \sin(6x^2)\,dx$.

I had to use a website to find the seven derivatives for the polynomial, because it would have taken me hours to solve it. Then I did the integral of the seven polynomials and I got the correct answer of $2x^3-\frac{(36x^7)}{7}$.

After all that, for the second part, I used the fundamental theorem of calculus (eg. $F(0.75)-F(0)$) to find the correct answer of 4509/28672.

My question is: How do you solve this quicker? This would have taken at least a couple of hours to solve without a computer to get the seven derivatives when doing $\frac{f^n(a)}{n!}$ to get the polynomials.

Answer 1

Hint:

Find the MacLaurin polynomial of $\sin 6x^2$ at order $6$ and integrate it.