How to find indefinite integral of $\sin(x^3)$

Question

I obtained$$\int\sin(x^3)dx=\frac{\cos(x^3)}{3x^2}$$ was my answer which is wrong.

Is this somehow related to the indefinite integral of $\frac{f'(x)}{f(x)}$ being $\ln(f(x))$?

Answer 1

Probably, one of the simplest things you could do is to expand the integrand as an infinite series using $$\sin(y)= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} y^{2n+1}$$ making $$\sin(x^3)= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{6n+3}$$ $$\int \sin(x^3)\,dx=\sum^{\infty}_{n=0}\frac{(-1)^n}{(6n+4)(2n+1)!} x^{6n+4}$$ Otherwise, you will face quite complex functions.