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I'd like to ask if someone can please give me a little push with this assignment:

Approximate the value of the integral $\int_0^1 \sin(x^2) dx$ using only $\mathbb{N}$ numbers and basic operations $(+,-,*,/)$. For example use Simpson's rule or the Trapezoidal rule to calculate the integral, then use Taylor's series to determine the value of sinus.

I have used Simpson's rule to approximate the integral and got:

$\int_0^1 \sin(x^2) dx \approx \frac{2}{3}\sin(\frac{1}{4}) + \frac{1}{6}\sin(1)$

But I don't know what to do with the Taylor series. Should I compute it for $\frac{2}{3}\sin(x^2) + \frac{1}{6}\sin(x^2)$ and then express it for $x = \frac{1}{4}$ and $x = 1$ or is my thinking bad?

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But I don't know what to do with the Taylor series.

What you do here is to figure out the Maclaurin expansion for the function $\int_0^x \sin(u^2)\,\mathrm du$. You can figure that by starting with the Maclaurin series for $\sin\,x$ and then replacing the $x$ with $x^2$. Use that series when you're called to evaluate $\sin(x^2)$ for trapezoidal or Simpson's purposes.

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    Sorry, I didn't want to put words in your mouth, it's just that the second part o$f$ the assigment is to try to integrate the Taylor $p$olynonmial itsel$f$, so I got a little bit mixed up about your suggestions. So once again - sorry!2011-10-29