Integral de log10(x)*(x+10)^(-1) dx
Solución
Respuesta (Indefinida)
[src]
/ / x \
| - polylog|2, 1 + --| + log(10)*log(10 + x) + 3*pi*I*log(10 + x) for |10 + x| < 1
| \ 10/
|
| / x \ / 1 \ / 1 \ 1
< - polylog|2, 1 + --| - log(10)*log|------| - 3*pi*I*log|------| for -------- < 1
/ | \ 10/ \10 + x/ \10 + x/ |10 + x|
| |
| / log(x)\ | / x \ __0, 2 /1, 1 | \ __2, 0 / 1, 1 | \ __2, 0 / 1, 1 | \ __0, 2 /1, 1 | \
| |-------| |- polylog|2, 1 + --| + log(10)*/__ | | 10 + x| - log(10)*/__ | | 10 + x| - 3*pi*I*/__ | | 10 + x| + 3*pi*I*/__ | | 10 + x| otherwise
| \log(10)/ \ \ 10/ \_|2, 2 \ 0, 0 | / \_|2, 2 \0, 0 | / \_|2, 2 \0, 0 | / \_|2, 2 \ 0, 0 | /
| --------- dx = C + -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| x + 10 log(10)
|
/
∫x+10log(10)1log(x)dx=C+log(10)⎩⎨⎧log(10)log(x+10)+3iπlog(x+10)−Li2(10x+1)−log(10)log(x+101)−3iπlog(x+101)−Li2(10x+1)−G2,22,0(0,01,1x+10)log(10)−3iπG2,22,0(0,01,1x+10)+G2,20,2(1,10,0x+10)log(10)+3iπG2,20,2(1,10,0x+10)−Li2(10x+1)for∣x+10∣<1for∣x+10∣1<1otherwise
Gráfica
2
2 pi
log (10) - --- + 3*pi*I*log(10)
-polylog(2, 7/2) + log(10)*log(35) + 3*pi*I*log(35) 6
--------------------------------------------------- - -------------------------------
log(10) log(10)
−log(10)−6π2+log(10)2+3iπlog(10)+log(10)log(10)log(35)−Li2(27)+3iπlog(35)
=
2
2 pi
log (10) - --- + 3*pi*I*log(10)
-polylog(2, 7/2) + log(10)*log(35) + 3*pi*I*log(35) 6
--------------------------------------------------- - -------------------------------
log(10) log(10)
−log(10)−6π2+log(10)2+3iπlog(10)+log(10)log(10)log(35)−Li2(27)+3iπlog(35)
(-polylog(2, 7/2) + log(10)*log(35) + 3*pi*i*log(35))/log(10) - (log(10)^2 - pi^2/6 + 3*pi*i*log(10))/log(10)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.