Does (E ∧ x) * ((SiNx) ∧ 2) have an original function? If so, what is it?

Does (E ∧ x) * ((SiNx) ∧ 2) have an original function? If so, what is it?

I = ∫ e ^ xcos2xdx = e ^ xcos2x + ∫ 2E ^ xsin2xdx = e ^ xcos2x + 2 [e ^ xsin2x - ∫ 2E ^ xcos2xdx] = e ^ xcos2x + 2E ^ xsin2x-4i: I = e ^ x (cos2x +...)

High number function problem, urgent, prove that x = SiNx has only one real root

The derivation of X and SiNx is 1, and the derivation of SiNx is cosx. On [0, π / 2], cosx is decreasing, that is, cosx

On a theorem in higher mathematics, we can split functions like SiNx
I remember there seemed to be such a way
We can decompose SiNx into polynomials about X. the degree of X in each term is different. I don't remember what this formula is.
Please answer.

Taylor expansion~
sinx=x-x^3/3!+x^5/5!-x^7/7!+x^9/9!-…… Here it is written in the form of infinite series

Solving primitive function of 1 / (1 + (SiNx) &# 178;) by indefinite integral of higher number