Question

部分分式积分法，求解几题积分题

1. (x+4)/(x^2+5x-6)
2. (x)/(x^2-2x-3)
忘了还有一题，(x^3)/(x^2+2x+1)

Answer 1

1.设 (x+4)/(x^2+5x-6)=a/(x-1)+b/(x+6),
则x+4=a(x+6)+b(x-1)=(a+b)x+6a-b,
比较系数得a+b=1,6a-b=4,
解得a=5/7,b=2/7.
∴∫ (x+4)/(x^2+5x-6)*dx=(5/7)ln(x-1)+(2/7)ln(x+6)+C.

2. 设x/(x^2-2x-3)=a/(x-3)+b(x+1),
则x=a(x+1)+b(x-3)=(a+b)x+a-3b,
比较系数得a+b=1,a-3b=0,
解得a=3/4,b=1/4.
∴∫xdx/(x^2-2x-3)=(3/4)ln(x-3)+(1/4)ln(x+1)+C.