Find integral approximately

Question

Find $\int_0^1\frac{\sin(nx)}{1+x}dx$ accurate within $O(\frac{1}{n^3})$. Please, can you give me a hint how to calculate this integral approximately? I know that $\sin(x) = x + o(x)$, but it is not enough for $O(\frac{1}{n^3})$. Can I use Taylor series?

Answer 1

The antiderivative of $\frac{\sin (n x)}{x+1}$ is: $$\int \frac{\sin (n x)}{x+1} \, dx = \cos (n)\, \text{Si}(n (x+1))-\sin (n)\, \text{Ci}(n (x+1))$$ which can be proven by differentiation. $\text{Si}(x)$ and $\text{Ci}(x)$ is the sine integral an cosine integral, respectively. Expanding the result for the definite integral in a Taylor series gives the requested result.