n趋向无穷 lim∫(0,1)e^-xsinnxdx怎么求？ limn→∞∫10e-xsinnxdx=______

\u6c42\u5b9a\u79ef\u5206[0,1]x^ne^xsinnxdx\u5728n\u8d8b\u5411\u65e0\u7a77\u65f6\u7684\u6781\u9650

\u628a\u79ef\u5206\u533a\u95f4\u5206\u4e3a\u4e24\u90e8\u5206[0,1/2]&[1/2,1]\u3002
\u7b2c\u4e00\u90e8\u5206\u4e2d\uff0c\u88ab\u79ef\u51fd\u6570\u4e00\u81f4\u6536\u655b\u4e8e0\uff0c\u4ece\u800clim\u4e0e\u222b\u8fd0\u7b97\u53ef\u4ee5\u4ea4\u6362\u6b21\u5e8f\uff0c\u5f97\u51fa\u503c\u4e3a0\uff1b
\u7b2c\u4e8c\u90e8\u5206\u4e2d\uff0c\u88ab\u79ef\u51fd\u6570\u8fd8\u6709sinnx\u8fd9\u4e00\u9879\uff0c\u4f59\u4e0b\u7684x^n*e^x\u5728[1/2,1]\u4e0a\u9ece\u66fc\u53ef\u79ef\uff0c\u8fd0\u7528Riemman-Lebesgue\u5f15\u7406\u4ea6\u53ef\u5f97\u6781\u9650\u4f4d0\uff1b
\u7efc\u5408\u4e24\u90e8\u5206\u7684\u7ed3\u679c\u53ef\u5f97\u6b64\u6781\u9650\u7b49\u4e8e0\u3002#

\u8bbeIn\uff1d\u222be?xsinnxdx\uff0c\u5219In=-\u222bsinnxde-x=-e-xsinnx+n\u222be-xcosnxdx=-e-xsinnx-n\u222bcosnxde-x=-e-xsinnx-ne-xcosnx-n2\u222be-xsinnxdx=-e-x\uff08sinnx+ncosnx\uff09-n2In\u2234In\uff1d?sinnx+ncosnxn2+1e?x+C\u2234\u222b10e?xsinnxdx\uff1d?ncosn+sinnn2+1e?1+nn2+1\u2234limn\u2192\u221e\u222b10e?xsinnxdx\uff1dlimn\u2192\u221e[?ncosn+sinnn2+1e?1+nn2+1]=0

∫(0,1)e^-xsinnxdx
=-∫(0,1)sinnxde^-x
=-sinnx e^(-x)|(0,1) +∫(0,1)e^(-x)dsinnx
= -sin(n)/e +n∫(0,1)e^(-x)cosnxdx
=- sin(n)/e -n∫(0,1) cosnx d e^(-x)
=-sin(n)/e -ncosnxe^(-x)|(0,1) + n∫(0,1) e^(-x) d cosnx
=-sin(n)/e - ncosn /e + n -n^2 ∫(0,1) e^(-x)sinnx d x
∫(0,1) e^(-x)sinnx d x = -(n+sin(n)/e+ncos(n)/e)/(1+n^2)
去极限后值为0

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