I have an assignment where I have to compute the following integral $$\int_1^{1.71}\frac {\cos \sqrt[4]{x}}{1+x\sqrt{x}}dx$$ to twelve correct significant digits.
What is the best method to achieve this when computation time is not a big issue? How do I know how many digits are correct?
On
Get a bound on the derivative on the domain. This in turn gives you a relationship with the Lipschitz constant L. This tells you how much point estimates can vary. So if you sample points in an interval I of size t, the error in calculating f(x) in I is at most Lt. Comment if more elaboration needed.
Note, this only works here as we can understand the derivative of your integrand.
On
The "best" simple method is Romberg integration. Let $M(n)$ denote the midpoint integration with $2^n$ intervals, $T(n)$ the trapezoidal integration and $S(n)$ the Simpson method. Then one can quickly check that $T(n)+M(n)=2T(n+1)$ and by Richardson extrapolation $$ S(n) = \frac{4T(n+1)-T(n)}3=\frac{T(n)+2M(n)}3 $$ The Simpson method has error order 4, so that the next step in the Romberg integration produces the numbers $\frac{16S(n+1)-S(n)}{15}$ with error order 6 and so on. $$\small \begin{array}{l|lllll} n& p=2&p=4&p=6&p=8&p=10\\\hline 0 & 0.1413610117349 \\ 1 & 0.1357347551451& 0.1338593362819 \\ 2 & 0.1343011488019& 0.1338232800209& 0.1338208762701 \\ 3 & 0.1339410393737& 0.1338210028976& 0.1338208510894& 0.1338208506897 \\ 4 & 0.1338509052773& 0.1338208605785& 0.1338208510906& 0.1338208510906& 0.1338208510922 \\ 5 & 0.1338283650834& 0.1338208516855& 0.1338208510926& 0.1338208510926& 0.1338208510926 \\ 6 & 0.1338227296181& 0.1338208511297& 0.1338208510926& 0.1338208510926& 0.1338208510926 \\ 7 & 0.1338213207258& 0.1338208510950& 0.1338208510926& 0.1338208510926& 0.1338208510926 \\ \end{array} $$
One can take the the next value down in the table as reference to compute an error estimate.
If you consider performances, you are correct with the problem of the square and fourth root.
Independently of the selected numerical method, I should let $x=t^4$ to make $$\int_1^{1.71}\frac {\cos \sqrt[4]{x}}{1+x\sqrt{x}}dx=4\int_1^{\sqrt[4]{1.71}}\frac{ t^3 }{t^6+1}\cos (t)\,dt$$ Now, you face a smaller integration range and the second integrand is much closer to linearity than the first one.
You could even use series expansion around $t=1$ to $O\left((t-1)^{n+1}\right)$ and it would converge quite fast $$\left( \begin{array}{cc} n & \text{result} \\ 0 & 0.13776738977500630815 \\ 1 & 0.13244166602060158949 \\ 2 & 0.13379101200456405488 \\ 3 & 0.13389678244323662667 \\ 4 & 0.13381286137858445987 \\ 5 & 0.13381750204286473434 \\ 6 & 0.13382176024778008614 \\ 7 & 0.13382093814245070512 \\ 8 & 0.13382078258061243158 \\ 9 & 0.13382085458575250985 \\ 10 & 0.13382085507917192293 \\ 11 & 0.13382085032932136977 \\ 12 & 0.13382085092985845912 \\ 13 & 0.13382085116591896269 \\ 14 & 0.13382085109375183538 \\ 15 & 0.13382085108748612023 \\ 16 & 0.13382085109326772813 \\ 17 & 0.13382085109291259286 \\ 18 & 0.13382085109256093758 \\ 19 & 0.13382085109263371344 \\ 20 & 0.13382085109264832238 \\ \cdots & \cdots \\ \infty &0.13382085109264158173 \end{array} \right)$$