Solution of integral

Do you mean, integrate x^x dx? If yes, Wolfram Alpha displays this.

Yes .

Function x^x
This function cannot be integrated.
Most likely it's a nonelementary integral. from http://www.numberempire.com/integralcalculator.php.

Mathematic gives this as the series expansion at x=0...

x+1/4 x^2 (2 log(x)-1)+1/54 x^3 (9 log^2(x)-6 log(x)+2)+1/768 x^4 (32 log^3(x)-24 log^2(x)+12 log(x)-3)+(x^5 (625 log^4(x)-500 log^3(x)+300 log^2(x)-120 log(x)+24))/75000+(x^6 (324 log^5(x)-270 log^4(x)+180 log^3(x)-90 log^2(x)+30 log(x)-5))/233280+(x^7 (117649 log^6(x)-100842 log^5(x)+72030 log^4(x)-41160 log^3(x)+17640 log^2(x)-5040 log(x)+720))/592950960+(x^8 (131072 log^7(x)-114688 log^6(x)+86016 log^5(x)-53760 log^4(x)+26880 log^3(x)-10080 log^2(x)+2520 log(x)-315))/5284823040+(x^9 (4782969 log^8(x)-4251528 log^7(x)+3306744 log^6(x)-2204496 log^5(x)+1224720 log^4(x)-544320 log^3(x)+181440 log^2(x)-40320 log(x)+4480))/1735643790720+(x^10 (1562500 log^9(x)-1406250 log^8(x)+1125000 log^7(x)-787500 log^6(x)+472500 log^5(x)-236250 log^4(x)+94500 log^3(x)-28350 log^2(x)+5670 log(x)-567))/5670000000000+x^11 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log^2(x)+12162150 log(x)-868725))/9653278129532887449600+(x^15 (4805419921875 log^14(x)-4485058593750 log^13(x)+3887050781250 log^12(x)-3109640625000 log^11(x)+2280403125000 log^10(x)-1520268750000 log^9(x)+912161250000 log^8(x)-486486000000 log^7(x)+227026800000 log^6(x)-90810720000 log^5(x)+30270240000 log^4(x)-8072064000 log^3(x)+1614412800 log^2(x)-215255040 log(x)+14350336))/6283924459312500000000000+(x^16 (562949953421312 log^15(x)-527765581332480 log^14(x)+461794883665920 log^13(x)-375208342978560 log^12(x)+281406257233920 log^11(x)-193466801848320 log^10(x)+120916751155200 log^9(x)-68015672524800 log^8(x)+34007836262400 log^7(x)-14878428364800 log^6(x)+5579410636800 log^5(x)-1743565824000 log^4(x)+435891456000 log^3(x)-81729648000 log^2(x)+10216206000 log(x)-638512875))/11778483592893497717293056000+x^17 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There are a whole bunch of terms that cannot be integrated, such as S e^(x^2) dx. If you have to work with them use the appropriate series.

all i can say is this.. O_O well its not words but AHHHHHHHHH!!!!!!!! wow neva seen so many numbers bunched up numbers like that in my LIFE!!!!!!!!!!

Where did you find that?