expansion or taylor

  1. G

    Integration by expansion

    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.
  2. G

    Integration

    Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) e^{xcos[\pi (t-1)/2]} dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) \end{equation} as $x\rightarrow0$. => Using integration by parts, but its too complicated for me because of huge exponential term...
  3. P

    Series Expansion/Taylor Series Help

    Okay, I am in dire need of help. How do I know that sin(x) is equal to x-((x^3)/6)+......+(-1)^n((x^(2n+1))/(2n+1)!)+x^(2n+1)E(x), (where E is epsilon)? Same goes for cos(x), arctan(x), and all the other functions that can be represented by the sum of a series(I think that's how you say...