判断广义积分的敛散性,若收敛计算其值 1 .∫[2,+∞]1/(1-x^2) dx 1 .∫[-∞,+∞]1/(x²+2x+2) dx
问题描述:
判断广义积分的敛散性,若收敛计算其值 1 .∫[2,+∞]1/(1-x^2) dx 1 .∫[-∞,+∞]1/(x²+2x+2) dx
答
∫[2,+∞]1/(1-x^2) dx =1/2∫[2,+∞][1/(1-x) -1/(1+x)]dx =-1/2∫[2,+∞][ 1/(1+x)-1/(x-1)]dx =-1/2[ln(1+x)-ln(x-1)][2,+∞]=-1/2ln[(1+x)/(x-1)][2,+∞]=1/2*ln3∫[-∞,+∞]1/(x²+2x+2) dx=∫[-∞,+∞...