求定积分，积分上下限0、+∞,被积函数xe^-x 高数问题，计算 积分xe^-(x+y)dx,上下限分别为+∞...
\u79ef\u5206\u9898\u222bxe^(-x)dx \u5728\uff080\uff0c+\u221e\uff09\u4e0a\u6c42\u5b9a\u79ef\u5206\u5bf9\u4e8e\u8fd9\u4e2a\u79ef\u5206\u5f0f\uff0cu=x,v'=e^(-x)\u6240\u4ee5u\u2018=1\uff0cv=-e^(-x)\u222bxe^(-x)dx=-xe^(-x)-\u222be^(-x)dx=-xe^(-x)+e^(-x)+C=(1-x)e^(-x)+C\u6240\u4ee5\u5728(0\uff0c+\u221e)\u79ef\u5206\u662f(1-x)e^(-x)+C
\u5982\u679c\u4e24\u79cd\u7b97\u6cd5\u7ed3\u679c\u4e0d\u4e00\u6837\uff0c\u603b\u6709\u4e00\u4e2a\u9519\u4e86\uff0c\u5173\u952e\u4f60\u6ca1\u8d34\u51fa\u4f60\u600e\u4e48\u7b97\u7684\uff0c\u8c01\u77e5\u5230\u4f60\u600e\u4e48\u9519\u8bef\u7684\uff0c\u96be\u9053\u4f60\u4ee5\u4e3a\u5b83\u5929\u7136\u5c31\u4e0d\u5e94\u8be5\u7b49\u4e8ee^(-y)?
\u222b\uff08x+y-y\uff09e^-(x+y)dx=\u222b\uff08x+y\uff09e^-(x+y)d(x+y) -\u222bye^-(x+y)dx
=-\u222b\uff08x+y\uff09de^-(x+y) -\u222bye^-(x+y)dx
=-\uff08x+y)e^-(x+y)|0,+\u221e +\u222be^-(x+y)d(x+y)-\u222bye^-(x+y)dx
=ye^(-y) -e^-(x+y)|0,+\u221e -\u222bye^-(x+y)dx
=ye^(-y) +e^(-y) +ye^-(x+y)|0,+\u221e
=ye^(-y) +e^(-y) -ye^(-y)
=+e^(-y)
\u770b\u6765\u53ea\u662f\u4f60\u7b97\u9519\u4e86
= -∫(0到+∞)x de^-x，分部积分法第一步
= -xe^-x + ∫(0到+∞)e^-x dx，分部积分法第二步
= -[lim(x->+∞)xe^-x - lim(x->0)xe^-x] - e^-x
= 0 - [lim(x->+∞)e^-x - lim(x->0)e^-x]
= -[0 - 1]
= 1
绛旓細鍘熷紡=-鈭玡^(-x/2000)d(-x/2000)=-e^(-x/2000)x鈫+鈭 鍒-x/2000鈫-鈭 鎵浠^(-x/2000)鈫0 鎵浠ュ師寮=0-(-e^0)=1
绛旓細鈭玿e^-x(y+1)dy=鈭玡^-x(y+1)dx(y+1)=-e^-x(y+1)|y=鏃犵┓-e^-x(y+1)|y=0 =0鈥攅^-x=-e^-x
绛旓細dx锛宼锛0 鎵浠 sint/t锛濃埆(0,+鈭) e^(-xt)sint dx 鈭(0,+鈭) sint/tdt 锛濃埆(0,+鈭) [鈭(0,+鈭) e^(-xt)sint dx] dt 浜ゆ崲绉垎娆″簭 锛濃埆(0,+鈭) [鈭(0,+鈭) e^(-xt)sint dt] dx 涓棿鐨绉垎姹鍑哄師鍑芥暟鍚庝唬瀹氱Н鍒鈭玸int/t dt 涓婇檺鏄棤绌锛屼笅闄鏄浂銆傛庝箞鍋氾紝
绛旓細鈭(x0->鏃犵┓) dx/x^2 棣栧厛鍒╃敤 鈭玠x/x^2 =-1/x +C =-[1/x]|(x0->鏃犵┓)浠ｅ叆瀹氱Н鍒鐨涓婁笅闄 =1/x0- lim(x0->鏃犵┓) 1/x 璁＄畻 lim(x0->鏃犵┓) 1/x =0 =1/x0 - 0 鍖栫畝 =1/x0 鎵浠ョ敱涓婅堪璁＄畻锛鍙緱 鈭(x0->鏃犵┓) dx/x^2 = 1/x0 ...
绛旓細棣栧厛澹版槑(a,b)鈭玣(x)dx锛岀Н鍒涓婇檺涓篵锛堝彸杈归偅涓級锛屼笅闄涓篴锛屽嵆a鍒癰锛屼笉瑕佺湅鍙嶄簡 妤间笂锛屾棤绌凤紞鏃犵┓鏄笉瀹氬瀷锛屾槸鍙兘鏀舵暃鐨勩3绉嶆柟娉曞仛杩欎釜棰 1.浜岄噸绉垎杞寲娉 绉垎鍙湅浣 F(y)=(0,+鈭)鈭玔e^(-蟺x)]/xdx F(1)-F(2)=(0,+鈭)鈭玔e^(-蟺x)-e^(-2蟺x)]/xdx f(y)=F'...
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绛旓細鏈鍚庣粨鏋滄槸姝ｆ棤绌 鍙傝冭祫鏂欙細http://www.wolframalpha.com/input/?i=Integrate%5Bx%2F%281%2Bx%5E2%29%5D
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