How Does Integration by Expansion Work for Sine Integrals?

[wel]

Consider the integral
\begin{equation}
I(x)= \frac{1}{\pi} \int^{\pi}_{0} sin(xsint) dt
\end{equation}
show that
\begin{equation}
I(x)= 4+ \frac{2x}{\pi}x +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.

(Note: It is not my homework or coursework question but it is from past exam paper which i am preparing for my exam)

 

[Simon Bridge]

Please show your working.

Maclearen series: $$f(t)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$$

$f(t)=\sin(x\sin t),\; f(0)=0$

$f'(t)=\cdots$

$f''(t)=\cdots$

etc. until you start getting terms in x³

 

[Mark44]

wel,
Physics Forums rules require that you show what you have tried. I sent you a PM about this, and am closing this thread. You are welcome to start a new thread for this problem, but you need to show what you have tried.