1樓:飄渺的綠夢
^方法一:
∫[x^3/(x^2+x-6)]dx
=∫{x^3/[(x+3)(x-2)]}dx
=(62616964757a686964616fe59b9ee7ad94313333353461371/5)∫[x^3/(x-2)-x^3/(x+3)]dx
=(1/5)∫[(x-2+2)^3/(x-2)]dx-(1/5)∫[(x+3-3)^3/(x+3)]dx
=(1/5)∫(x-2)^2dx+(6/5)∫x-2)dx+(12/5)∫dx+(8/5)∫[1/(x-2)]dx
-(1/5)∫(x+3)^2dx+(9/5)∫(x+3)dx-(27/5)∫dx+(27/5)∫[1/(x+3)]dx
=(1/15)(x-2)^3+(3/5)(x-2)^2+(12/5)x+(8/5)ln|x-2|
-(1/15)(x+3)^3+(9/10)(x+3)^2-(27/5)x+(27/5)ln|x+3|+c。
方法二:
∫[x^3/(x^2+x-6)]dx
=∫[(x^3+x^2-6x-x^2+6x)/(x^2+x-6)]dx
=∫xdx-∫[(x^2-6x)/(x^2+x-6)]dx
=(1/2)x^2-∫[(x^2+x-6-7x+6)/(x^2+x-6)]dx
=(1/2)x^2-∫dx+∫[(7x-6)/(x^2+x-6)]dx
=(1/2)x^2-x+7∫[x/(x^2+x-6)]dx-6∫[1/(x^2+x-6)]dx
=(1/2)x^2-x+(7/5)∫[x/(x-2)-x/(x+3)]dx
-(6/5)∫[1/(x-2)-1/(x+3)]dx
=(1/2)x^2-x+(7/5)∫dx+(14/5)∫[1/(x-2)]dx-(7/5)∫dx
+(21/5)∫[1/(x+3)]dx-(6/5)∫[1/(x-2)-1/(x+3)]dx
=(1/2)x^2-x+(8/5)∫[1/(x-2)]dx+3∫[1/(x+3)]dx
=(1/2)x^2-x+(8/5)ln|x-2|+(27/5)ln|x+3|+c。
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