∫(1到正无穷大)dx/{x^2(x+1)}

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∫(1到正无穷大)dx/{x^2(x+1)}

∫(1到正无穷大)dx/{x^2(x+1)}
∫(1到正无穷大)dx/{x^2(x+1)}

∫(1到正无穷大)dx/{x^2(x+1)}
1/x²(x+1)=(Ax+B)/x²+C/(x+1)
(Ax+B)(x+1)+Cx²=1
Ax²+Ax+Bx+B+Cx²=1
(A+C)x²+(A+B)x+B=1
A+C=0
A+B=0
B=1
A=-1
C=1
1/x²(x+1)= (-x+1)/x²+1/(x+1)
∫(1-->+∝)dx/{x²(x+1)}
=∫(1-->+∝)[(-x+1)/x²+1/(x+1)]dx
=∫(1-->+∝)[-1/x+1/x²+1/(x+1)]dx
=-lnx-1/x+ln(1+x) 1-->+∝
=-1/x+ln(1+x)/x
=1-ln2