# Representation of power series of product of sine and cosine

Given $$f(x)= \int \limits_0^x \sin(y^2) \cos(y^2) \mathrm{d}y$$

Anyone can help and guide me for this?I don't really have an idea of how to represent it as power series

Thank you!

My attempt: $$0.5\int_{0}^x \sum_{n=0}^\infty \frac{{(-1)}^n{({2{y}^{2})}}^{2n+1}}{(2n+1)!} \mathrm{d}y$$ after substituting Maclaurin series of sin x

• Why would you want to represent it (Who? Each of the functions or their integral?) as a power series? Commented Apr 19, 2015 at 10:49
• @Timbuc represent f(x) as power series Commented Apr 19, 2015 at 10:53
• Expand the integrand (after its simplification) as Taylor series and integrate. Commented Apr 19, 2015 at 11:39

Hint:

$$\int_0^x\sin t^2\cos t^2\;dt=\frac12\int_0^x\sin 2t^2\;dt$$

and then you may want to check Fresnel Integrals

• Can I check answer with you? [(-1)^n (2x^2)^(2n+2)]/2(2n+2)! Commented Apr 19, 2015 at 11:19
• interval x in (-inf,inf) Commented Apr 19, 2015 at 11:27
• @UnusualSkill You've been a member of this site for looooong time: if you want to be properly understood please do begin learning the easy directions to properly write mathematics in this site. I can't understand completely what you wrote, though I'm afraid it is wrong. Check the following : meta.math.stackexchange.com/questions/5020/… Commented Apr 19, 2015 at 11:31
• $$\sum_{n=0}^\infty \frac{{(-1)}^{n}{(2x^2)}^{2n+2}}{2(2n+1)!}$$ Commented Apr 19, 2015 at 11:43
• It does not seem to be correct. Check your edit. Commented Apr 19, 2015 at 11:45

Hint: use $\sin 2u = 2 \sin u \cos u$; the power series of $\sin$ and term by term integration.

• Can I check answer with you? [(-1)^n (2x^2)^(2n+2)]/2(2n+2)! Commented Apr 19, 2015 at 11:18
• interval x in (-inf,inf) Commented Apr 19, 2015 at 11:27
• it seems fine. edit your question with your attempt! Commented Apr 19, 2015 at 12:12
• really?becuz other said it is not right... Commented Apr 19, 2015 at 12:39
• my important attempt Commented Apr 19, 2015 at 12:58