Integral approximation via Taylor series

[NYC]

In terms of Taylor series, there is one final question which I'm not sure how to approach. I have been asked to integrate (cos(sqrt x))/x from one to two, finding an answer accurate to two decimal places. My best guess is to attempt a Taylor series centered at a=1, but I'm still confused as to how to do this one. In case my writing isn't clear, for which I apologize, the problem reads "the integral of the cosine of the square root of x ALL divided by x...evaluated from one to two." My problem arises in establishing a pattern in the derivatives, but any help would be appreciated.

 

[galactus]

Start with the series for cos(x) and use \(\displaystyle \sqrt{x}\) instead of x.

Use the MacLaurin, where a=0.

\(\displaystyle \L\\1-\frac{x}{2!}+\frac{x^{2}}{4!}-\frac{x^{3}}{6!}+\frac{x^{4}}{8!}-\frac{x^{5}}{10!}\)

Divide it by x:

\(\displaystyle \L\\\frac{-1}{2!}+\frac{1}{x}+\frac{x}{4!}-\frac{x^{2}}{6!}+\frac{x^{3}}{8!}-\frac{x^{4}}{10!}\)

Now integrate:

\(\displaystyle \L\\\frac{-x}{2}+\frac{x^{2}}{2(2!)}-\frac{x^{3}}{3(6!)}+\frac{x^{4}}{4(8!)}-\frac{x^{5}}{5(10!)}-ln(x)\)

Now, sub in x=2 and x=1, subtract. You should get an answer more accurate than 2 decimal places using this much of the series.