Integral and MacLaurin polynomial problem

Question

I can't seem to get this problem: So far I obtained the polynomial:

But it is wrong.

Thank you in advance for the help!

Answer 1

We have that $$\sin(t) = t-\frac{t^3}{3!}+\frac{t^5}{5!}+\ldots$$

Substitute $3t^2$ in for $t$ in the above equation, and evaluate the integral.

Answer 2

If you do what Hossmeister suggested in his/her answer, you have $$\sin(3t^2)=3 t^2-\frac{9 }{2}t^6+O\left(t^{10}\right)$$ so $$\int \sin(3t^2)\,dt=t^3-\frac{9 t^7}{14}+O\left(t^{11}\right)$$ $$\int_0^x \sin(3t^2)\,dt=x^3-\frac{9 }{14}x^7$$