sinx/x和(sin(x^2))/x^2在零到正无穷上的对应的广义积分值相等,如何论证?

德尔芬娜2022-10-04 11:39:541条回答

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kittyleewan112 共回答了18个问题 | 采纳率94.4%: (0,+∞)∫(sinx/x)^2dx=(1/2)*(0,+∞)∫(1-cos2x)/x^2dx
=(1/2)*(0,+∞)∫1/x^2dx-(1/2)*(0,+∞)∫cos2x/x^2dx
=(1/2)*(0,+∞)∫1/x^2dx-(1/2)*(0,+∞)∫cos2x/d(1/x)
=(1/2)*(0,+∞)(-1/x)-(1/2)*(0,+∞)cos2x/x+(1/2)*(0,+∞)∫sin2x/xdx
=(1/2)*(0,+∞)(cos2x-1)/x+(0,+∞)∫sint/tdt
=(0,+∞)∫sinx/xdx; 1年前