# Solving such integrals numerically?

1. Apr 11, 2005

### TSN79

Is there any way of solving $$\int_{0}^{1} \frac{1}{\sqrt{100-x^5}} dx$$ by some regular method? If not, how does one go about solving such integrals numerically?

2. Apr 11, 2005

### arildno

1. I don't know about any method to crack this integral analytically.
2. As for numerical methods:
These typically start with some fancy finite Riemann sum to approximate the integral.
You could check up on the Trapezoid rule or Simpson's rule or some other rule..

3. Apr 11, 2005

### dextercioby

$$\int_{0}^{1} \frac{dx}{\sqrt{100-x^{5}}} =\frac{1}{10} \ _{2}F_{1}\left(\frac{1}{2},\frac{1}{5},\frac{6}{5},\frac{1}{100}\right)$$

according to my friend,Maple.

Daniel.

4. Apr 11, 2005

### Zurtex

According to Mathematica:

$$\int_{0}^{1} \frac{dx}{\sqrt{100-x^{5}}} = \frac{1}{10} \text{Hypergeometric2F1} \left[ \frac{1}{5},\frac{1}{2},\frac{6}{5},\frac{1}{100} \right]$$

Hmm wonder why the fractions are different, does it make a difference?

Anyway to 100 s.f:

0.1000836762086654607634453194543998535914445494139945308954127536431450557575506273354667189780577481

Last edited: Apr 11, 2005
5. Apr 11, 2005

### dextercioby

Conventions,on normal basis they should be the same...

Gauss's function is $_{2}F_{1}\left(a,b,c,z\right)$

Daniel.

P.S.Here are 125 sig.digs.

$$\allowbreak .\,10008\,36762\,08665\,46076\,34453\,19454\,39985\,35914\,44549\,41399\,45308\,95412\,$$
$$75364\,31450\,55757\,55062\,73354\,66718\,97805\,77480\,65696\,66047\,63945\,64052\,7984$$

Last edited: Apr 11, 2005
6. Apr 11, 2005

### Zurtex

Ahh so it's going to be like that is it, well in that case, to 2000 sf:

0.1000836762086654607634453194543998535914445494139945308954127536431450557575\
506273354667189780577480656966604763945640527983993080671953023089297911515248\
996013282637274784238797044064349870623646876674724483462693995830072965962405\
190019049666775160232759330205668592578946075338958508523904278007888848833088\
060488556739935166104773509141768174230265551654691550416849059337448757791416\
602630567642386763728600302390370505747803022415391215722818064179823297750442\
434640467815965016423148168846638958221953928239081869470007493153876431418021\
157217678724106639004941740270189408736496188721513628469894013953107138449253\
791957534610058071313307798195976366400486448615041008866156544661543944553681\
807447545511569915551726649431123256963503190869480487316815727122678140209122\
738110692425545202808759680890972292418663402273431929145662980727173557540648\
152030821410027920261779677495993071187486068413623093242779260125263946810946\
539669010663042017697120941744353678783323221747296966263595416822115595315497\
544773344894826763667430935304194135002414853743041716586627006645641572144003\
038754961014790401073888263046103372397829209944018459246323194799313562020937\
625258141676770207971455031095503662945181184143286220172118052176868976940819\
363006570343110478540840140496393128738996412270357183445395020573388989278555\
813556144036563102455032850874625946333692746942852322508699926140544281120662\
532525325402041764035161169228790566576239939498698256797892476673581793593341\
565591033055741801465791277510653241163913554713152060127185501516632153831744\
084485276203982058024033779697637618294711132918471464969716668181939944429129\
590221571580066039538720668949128150132064065578982106579514989803518960288881\
581003421562857862205496312407174042287815539537348909650284394685560662915042\
685107335399595953096758002153118745564316408031460221673554730734019949375781\
303013799317984319741189620338837708872894884720821818310366605811188532339923\
2084126619627384173536035029500741646467671020765461

:tongue: :tongue: :tongue:

7. Apr 11, 2005

### dextercioby

Heh,i use a 10 y.o.Maple.It can't more than 1000 decimals.

Daniel.

8. Apr 11, 2005

### marlon

It always breaks my heart if an integral has no analytical solution. I hate it when those damned Maple or Mathematica-programs are needed in order to crack the problem. Doesn't it disturbe you that we don't have a TOE when it come to integrals. You know, how about a String Theory-variant for integral-solving...i dunno, just wondering...

regards
marlon

9. Apr 11, 2005

### dextercioby

I'm not pissed off.I like when the results come out in terms of "fancy functions" (i call them:"common special functions").

This is not really elementary mathematics...

Daniel.