求不定积分 (1) ∫xe^-xdx (2) ∫x^3lnxdx (3) ∫xln(x+1)dx

求不定积分 (1) ∫xe^-xdx (2) ∫x^3lnxdx (3) ∫xln(x+1)dx
求不定积分 (1) ∫xe^-xdx (2) ∫x^3lnxdx (3) ∫xln(x+1)dx

求不定积分 (1) ∫xe^-xdx (2) ∫x^3lnxdx (3) ∫xln(x+1)dx
(1)
∫xe^-x dx=-∫x d(e^-x)
=-xe^(-x)+∫e^-x dx
=-xe^(-x)-e^(-x)+C
=-(x+1)e^(-x)+C
(2)
∫x³lnx dx
=∫lnx d(x⁴/4)
=(1/4)x⁴lnx-(1/4)∫x⁴d(lnx)
=(1/4)x⁴lnx-(1/4)∫x³ dx
=(1/4)x⁴lnx-(1/4)*x⁴/4+C
=(1/4)x⁴lnx-(1/16)x⁴+C
=(1/16)x⁴(4lnx-1)+C
(3)
设u=x+1,x=u-1,dx=du
∫xln(x+1) dx
=∫(u-1)lnu du
=∫ulnu du-∫lnu du
=∫lnu d(u²/2)-(ulnu-∫u dlnu)
=(1/2)u²lnu-(1/2)∫u² dlnu-(ulnu-∫du)
=(1/2)u²lnu-(1/2)∫u du-(ulnu-1)
=(1/2)u²lnu-(1/4)u²-ulnu+C
=(1/2)(x+1)²ln(x+1)-(1/4)(x+1)²-(x+1)ln(x+1)+C
=(1/2)(x²-1)ln(x+1)-(1/4)x(x-2)+C

1, ∫xe^-xdx=-∫xde^(-x)=xe^(-x)-∫e^(-x)dx=-[xe^(-x)+e^(-x)]+c=-e^(-x)(x+1)+c
2, ∫x^3lnxdx=[∫lnxdx⁴]/4=[x⁴lnx-∫x⁴dlnx]/4=[x⁴lnx-x⁴/4]/4+c=[x⁴(lnx-1/4)]/4+c
3...

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1, ∫xe^-xdx=-∫xde^(-x)=xe^(-x)-∫e^(-x)dx=-[xe^(-x)+e^(-x)]+c=-e^(-x)(x+1)+c
2, ∫x^3lnxdx=[∫lnxdx⁴]/4=[x⁴lnx-∫x⁴dlnx]/4=[x⁴lnx-x⁴/4]/4+c=[x⁴(lnx-1/4)]/4+c
3, ∫xln(x+1)dx =[∫ln(x+1)dx²]/2=[x²ln(x+1)-∫x²dln(x+1)]/2==[x²ln(x+1)-∫x²/(x+1)dx)]/2
=[x²ln(x+1)-∫(x²-1+1)/(x+1)dx]/2=[x²ln(x+1)-∫(x-1)dx-∫1/(x+1)dx]/2
=[x²ln(x+1)-x²/2+x-ln(x+1)]/2+c

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