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求1/(x^3+x^2+x+1)的不定积分
题目详情
求1/(x^3+x^2+x+1)的不定积分
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答案和解析
∫1/(x^3+x^2+x+1)dx
=∫1/(x^2(x+1)+(x+1))dx
=∫1/((x+1)(x^2+1))dx
=∫A/(x+1)+(Bx+C)/(x^2+1)dx
Ax^2+A+Bx^2+Cx+Bx+C=1
A+B=0 C+B=0 A+C=1
A=1/2 B=-1/2 C=1/2
=1/2∫1/(x+1)+(-x+1)/(x^2+1)dx
=1/2∫1/(x+1)dx-1/2∫x/(x^2+1)dx+1/2∫1/(x^2+1)dx
=1/2(ln(x+1)-1/2ln(x^2+1)+arctg(x))+C
=∫1/(x^2(x+1)+(x+1))dx
=∫1/((x+1)(x^2+1))dx
=∫A/(x+1)+(Bx+C)/(x^2+1)dx
Ax^2+A+Bx^2+Cx+Bx+C=1
A+B=0 C+B=0 A+C=1
A=1/2 B=-1/2 C=1/2
=1/2∫1/(x+1)+(-x+1)/(x^2+1)dx
=1/2∫1/(x+1)dx-1/2∫x/(x^2+1)dx+1/2∫1/(x^2+1)dx
=1/2(ln(x+1)-1/2ln(x^2+1)+arctg(x))+C
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