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Use Taylor polynomials to approximate the integral of sin x / x

by RoRi

Using the Taylor polynomial approximation of \sin x find an approximation for the integral

    \[ \int_0^1 \frac{\sin x}{x} \, dx. \]

Give an estimate for the error of the approximation. [Define \frac{\sin x}{x} = 1 when x = 0.]

We know (from this exercise, Section 7.8, Exercise #1) that the Taylor polynomial approximation of \sin x is given by

    \[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} + E_{2n}(x), \qquad \text{where} \quad |E_{2n}(x)| \leq \frac{|x|^{2n+1}}{(2n+1)!}. \]

This implies

    \begin{align*} &&\frac{\sin x}{x} &= 1 - \frac{x^2}{3!} + \frac{x^4}{5!} + \frac{E_{2n}(x)}{x} \\[9pt] \implies && \int_0^1 \frac{\sin x}{x} \, dx &= \int_0^1 \left( 1 - \frac{x^2}{3!} + \frac{x^4}{5!} + \frac{E_{2n}(x)}{x} \right) \, dx \\[9pt] &&&= \left( x - \frac{x^3}{3\cdot 3!} + \frac{x^5}{5 \cdot 5!} \right) \Bigr \rvert_0^1 + \int_0^1 \frac{E_{2n}(x)}{x} \, dx \\[9pt] &&&= 0.9461 + R \end{align*}

where

    \[ |R| \leq \int_0^1 \frac{x^6}{7!} \, dx = \frac{1}{7\cdot 7!} \approx 0.00002. \]

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