定积分在0-pai/2上在什么函数中sinx和cosx可以互换？ 定积分中sinx cosx 在什么条件下可以互换？
\u4e3a\u4ec0\u4e48\u57280\u5230pi/2\u533a\u95f4\u79ef\u5206\u65f6\uff0csinx\u548ccosx\u53ef\u4ee5\u4e92\u6362\uff1f\u4ee4sinx-cosx=a
\u5219a^2=(sinx)^2+(cosx)^2-2sinxcosx=1-2sinxcosx
\u6240\u4ee5sinxcosx=(1-a^2)/2
\u6240\u4ee5y=-a+(1-a^2)/2=-a^2/2-a+1/2
a=sinx-cosx=\u221a2(sinx-\u03c0/4)
\u6240\u4ee5-\u221a2<=a<=\u221a2
y=-1/2(a+1)^2+1
\u5bf9\u79f0\u8f74a=-1\u3002\u5f00\u53e3\u5411\u4e0b
-\u221a2<=a<=\u221a2
\u6240\u4ee5a=-1,y\u6709\u6700\u5927\u503c=1
\u221a2\u6bd4-\u221a2\u79bb\u5bf9\u79f0\u8f74\u66f4\u8fdc\uff0c\u6240\u4ee5a=\u221a2\u65f6\uff0cy\u6709\u6700\u5c0f\u503c=-\u221a2-1/2
\u6240\u4ee5\u503c\u57df[-\u221a2-1/2\uff0c1]
\u4e0d\u662f\u4e92\u6362\uff0c\u662f\u6362\u5143t=\u03c0/2-u\u7136\u540e\u63a8\u7406\u7b49\u53f7\u6210\u7acb\u3002
\u5728\u4efb\u4f55\u51fd\u6570\u4e2d\uff0csinx\u548ccosx\u90fd\u53ef\u4ee5\u4e92\u6362\uff0c\u4f46\u524d\u63d0\u662f\u88ab\u79ef\u51fd\u6570\u4e2d\uff0c\u4e0d\u80fd\u6709\u9664\u4e86sinx\u548ccosx\u4e4b\u5916\u7684\u5176\u4ed6\u5f62\u5f0f\u7684\u81ea\u53d8\u91cf\uff0c\u5373\u222b(0,\u03c0/2) f(sinx) dx=\u222b(0,\u03c0/2) f(cosx) dx\u3002
\u222b\u220f (\u79ef\u5206\u4e0a\u9650) 0\uff08\u79ef\u5206\u4e0b\u9650\uff09dx \u222bsinx (\u79ef\u5206\u4e0a\u9650) 0\uff08\u79ef\u5206\u4e0b\u9650\uff09 f(x,y) dy \u5728\u4ea4\u6362\u79ef\u5206\u6b21\u5e8f\u662f\u4e0d\u662f\u7b49\u4e8e\u222b\u220f /2(\u79ef\u5206\u4e0a\u9650) 0\uff08\u79ef\u5206\u4e0b\u9650\uff09dy \u222b(\u220f-arcsiny)(\u79ef\u5206\u4e0a\u9650) arcsiny\uff08\u79ef\u5206\u4e0b\u9650\uff09 f(x,y) dx\uff0c\u753b\u4e2a\u79ef\u5206\u533a\u57df\u56fe\u5c31\u5f88\u663e\u7136\u4e86\u3002
\u5b9a\u79ef\u5206
\u5b9a\u79ef\u5206\u7684\u6b63\u5f0f\u540d\u79f0\u662f\u9ece\u66fc\u79ef\u5206\u3002\u7528\u9ece\u66fc\u81ea\u5df1\u7684\u8bdd\u6765\u8bf4\uff0c\u5c31\u662f\u628a\u76f4\u89d2\u5750\u6807\u7cfb\u4e0a\u7684\u51fd\u6570\u7684\u56fe\u8c61\u7528\u5e73\u884c\u4e8ey\u8f74\u7684\u76f4\u7ebf\u628a\u5176\u5206\u5272\u6210\u65e0\u6570\u4e2a\u77e9\u5f62\uff0c\u7136\u540e\u628a\u67d0\u4e2a\u533a\u95f4[a,b]\u4e0a\u7684\u77e9\u5f62\u7d2f\u52a0\u8d77\u6765\uff0c\u6240\u5f97\u5230\u7684\u5c31\u662f\u8fd9\u4e2a\u51fd\u6570\u7684\u56fe\u8c61\u5728\u533a\u95f4[a,b]\u7684\u9762\u79ef\u3002\u5b9e\u9645\u4e0a\uff0c\u5b9a\u79ef\u5206\u7684\u4e0a\u4e0b\u9650\u5c31\u662f\u533a\u95f4\u7684\u4e24\u4e2a\u7aef\u70b9a,b\u3002
即∫(0,π/2) f(sinx) dx=∫(0,π/2) f(cosx) dx
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