xln（1+x^3)的积分是
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∫xln(1+x³)dx
=(1/2)x²ln(1+x³)-(1/2)∫x² {1/(1+x³)} 3x²dx
=(1/2)x²ln(1+x³)-(3/2)∫x^4 {1/(1+x³)} dx
=(1/2)x²ln(1+x³)-(3/2)∫{x- x/(1+x³)} dx
=(1/2)x²ln(1+x³)-(3/4)x²+3/2∫ x/(1+x³)} dx
=(1/2)x²ln(1+x³)-(3/4)x²+1/2∫ {[-1/(1+x)]+(x+1)/(x²-x+1)} dx
=(1/2)x²ln(1+x³)-(3/4)x²-1/2ln(x+1)+∫ {[(x-1/2)/((x-1/2)²+3/4)]+(1/2)/((x-1/2)²+3/4)}dx
=(1/2)x²ln(1+x³)-(3/4)x²-1/2ln(x+1)+1/2l n[(x-1/2)²+3/4)]+√3arctan2(x-1/2)/√3+c
=(1/2)x²ln(1+x³)-(1/2)∫x² {1/(1+x³)} 3x²dx
=(1/2)x²ln(1+x³)-(3/2)∫x^4 {1/(1+x³)} dx
=(1/2)x²ln(1+x³)-(3/2)∫{x- x/(1+x³)} dx
=(1/2)x²ln(1+x³)-(3/4)x²+3/2∫ x/(1+x³)} dx
=(1/2)x²ln(1+x³)-(3/4)x²+1/2∫ {[-1/(1+x)]+(x+1)/(x²-x+1)} dx
=(1/2)x²ln(1+x³)-(3/4)x²-1/2ln(x+1)+∫ {[(x-1/2)/((x-1/2)²+3/4)]+(1/2)/((x-1/2)²+3/4)}dx
=(1/2)x²ln(1+x³)-(3/4)x²-1/2ln(x+1)+1/2l n[(x-1/2)²+3/4)]+√3arctan2(x-1/2)/√3+c
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