Question

Express the integral ∫sin(x^2)dx as a power series.

Answer 1

wouldnt we adapt sin(x^2) into a power series; and then intergrate that?

Answer 2

f0(x) = sin(a^2)
f1(x) = 2a cos(a^2)
f2(x) = -4a^2 sin(a^2)/2!
f3(x) = -8a^3 cos(a^2)/3!
f4(x) = 16a^4 sin(a^2)/4!
......
fn(x) = \(\cfrac{(-1)^n}{n!}\) something .... :)

Answer 3

might have to split it up into a sin series plus a cosine series to get the switching signs to go right

Answer 4

I'm not really understanding why sin(x^2) comes out to the answer you have

Answer 5

if i recall it correctly; i power series is a way to define a function as a polynomial that has the same "curve" of the given function... and we can define curves thru successive derivatives.

Answer 6

and depending on what the base point is will determine the values of the "coefficients" in the power series ...

Answer 7

Oh now I see.

Answer 8

this is a visual that shows how each successive polynomial of a power series can wrap itself to a cosine function at a base point of a=0

Answer 9

Thanks for the help