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∫x+3/x^2-x-2dx=?
题目详情
∫x+3/x^2-x-2dx=?
▼优质解答
答案和解析
(x+3)/(x²-x-2)
=(x+3)/(x+1)(x-2)
令A/(x+1)+B/(x-2)=(x+3)/(x+1)(x-2)
即(A(x-2)+B(x+1))/(x+1)(x-2)=(x+3)/(x+1)(x-2)
∴A+B=1 B-2A=3
解得A=-2/3 B=5/3
∴∫(x+3)/(x²-x-2)dx
=-2/3*∫1/(x+1)dx+5/3*∫1/(x-2)dx
=-2/3ln|x+1|+5/3ln|x-2|+C
=(5ln|x-2|-2ln|x+1|)/3+C
=(x+3)/(x+1)(x-2)
令A/(x+1)+B/(x-2)=(x+3)/(x+1)(x-2)
即(A(x-2)+B(x+1))/(x+1)(x-2)=(x+3)/(x+1)(x-2)
∴A+B=1 B-2A=3
解得A=-2/3 B=5/3
∴∫(x+3)/(x²-x-2)dx
=-2/3*∫1/(x+1)dx+5/3*∫1/(x-2)dx
=-2/3ln|x+1|+5/3ln|x-2|+C
=(5ln|x-2|-2ln|x+1|)/3+C
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