Finding the first-order term in the Maclaurin expansion of an integral

[ra_forever8]

Consider the integral
\begin{equation}
I(x)= \frac{1}{\pi} \int^{\pi}_{0} sin(xsint) dt
\end{equation}
show that
\begin{equation}
I(x)= \frac{2x}{\pi} +O(x^{3})
\end{equation}
as $x\rightarrow0$.
=> I Have used the expansion of McLaurin series of $I(x)$ but did not work.
please help me.

 

[stainburg]

I'm not sure if the expansion correct. However, we can do it as follow.

By McLaurin expansion, we have

$$f(x,t)=\sin(x\sin t)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^{2n-1}}{(2n-1)!}(\sin t)^{2n-1}=\sum_{n=1}^{\infty} f_n(x,t)$$.

Since we integrate $$f(x,t)$$ on $$E=[0,\pi]\ni t$$, we should test uniform convergence of the series, $$\sum_{n=1}^{\infty} f_n(x,t)$$, at first.

Clearly, $$|f_n(x,t)|\leq \frac{|x|^{2n-1}}{(2n-1)!}$$ on $$E\ni t$$, and $$\sum_{n=1}^{\infty} \frac{|x|^{2n-1}}{(2n-1)!}$$ converges uniformly on $$\mathbb{R}\ni x$$. Thus, $$\sum_{n=1}^{\infty} f_n(x,t)$$ converges uniformly on $$E\ni t$$.Hence $$\frac{1}{\pi}\int_E f(x,t)dt=\frac{1}{\pi}\int_E \sum_{n=1}^{\infty}f_n(x,t)dt=\frac{1}{\pi}\sum_{n=1}^{\infty}\int_E f_n(x,t)dt=\frac{1}{\pi}\int_E x\sin tdt+O(x^3)$$, as $$x$$ goes to $$0$$.

I think you can complete the rest.