Monday, 2 March 2015

calculus - Evaluate intfracpi20fracx2sinxdx



I want to evaluate π20x2sinxdx

First,I tried to evaluate like this:
π20x2sinxdx=π20x2(1+cosxsinx)dx1+cosx=π20x2(1+cosxsinx)d(sinx1+cosx)



=π20x2dlog(sinx1+cosx)=x2log(sinx1+cosx)|π202π20xlog(sinx1+cosx)dx



=0+2π20xlog(1+cosxsinx)dx=2π20xlog(1+cosx)dx2π20xlog(sinx)dx
=2π20xlogcot(x2)dx=8π40xlogcotxdx
but I can't proceed next step,help me,thanks.


Answer



At the price of special functions, the antiderivative could be computed

I=x2sinxdx=4ixLi2(eix)+ixLi2(e2ix)+4Li3(eix)12Li3(e2ix)2x2tanh1(eix) where appear the polylogarithm functions.



lim as given by Wolfram Alpha. This evaluates a \approx 1.54798.



For a fast approximation, we could use the superb approximation \sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi) which was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician (have a look here).
This would make
I \approx J= - \int \left(\frac{x^2}{4}+\frac{5 \pi ^3}{16 (x-\pi )}+\frac{5 \pi ^2}{16} \right)\,dx=-\frac{x^3}{12}-\frac{5 \pi ^2 x}{16}-\frac{5}{16} \pi ^3 \log (\pi -x)+\frac{19 \pi ^3}{48}

\lim_{x\to \frac{\pi }{2}} \, J=\frac{\pi ^3}{48} \left(11-15 \log \left(\frac{\pi }{2}\right)\right)\qquad \text{and} \qquad\lim_{x\to 0} \, J =\frac{\pi ^3}{48} (19-15 \log (\pi )) leading to the approximation
\int_{0}^{\frac{\pi}{2}}\frac{x^2}{ \sin x}dx\approx \frac{\pi ^3}{48} (15 \log (2)-8)\approx 1.54851 which is not too bad.



Tha advantage of such approximation is that it allows a fast evaluation of
K(t)=\int_{0}^{t}\frac{x^2}{ \sin x}dx The table below compares the approximation to the exact result
\left( \begin{array}{ccc} t & \text{approximation} & \text{exact} \\ \frac{\pi }{20} & 0.01221 & 0.01236 \\ \frac{\pi }{10} & 0.04936 & 0.04976 \\ \frac{3 \pi }{20} & 0.11258 & 0.11312 \\ \frac{\pi }{5} & 0.20358 & 0.20409 \\ \frac{\pi }{4} & 0.32475 & 0.32508 \\ \frac{3 \pi }{10} & 0.47939 & 0.47945 \\ \frac{7 \pi }{20} & 0.67196 & 0.67176 \\ \frac{2 \pi }{5} & 0.90847 & 0.90807 \\ \frac{9 \pi }{20} & 1.19701 & 1.19650 \\ \frac{\pi }{2} & 1.54851 & 1.54798 \\ \frac{11 \pi }{20} & 1.97802 & 1.97746 \\ \frac{3 \pi }{5} & 2.50657 & 2.50583 \\ \frac{13 \pi }{20} & 3.16447 & 3.16315 \\ \frac{7 \pi }{10} & 3.99696 & 3.99445 \\ \frac{3 \pi }{4} & 5.07529 & 5.07091 \\ \frac{4 \pi }{5} & 6.52008 & 6.51359 \\ \frac{17 \pi }{20} & 8.55922 & 8.55230 \\ \frac{9 \pi }{10} & 11.7067 & 11.7077 \\ \frac{19 \pi }{20} & 17.6067 & 17.6510 \end{array} \right)



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