x在0到正无穷,求x*e^(-x/y)在该范围的微积分
\u5fae\u79ef\u5206\u95ee\u9898\uff0c\u6b64\u9898X\u7684\u8303\u56f4\u4e3a\u4ec0\u4e48\u662f0\u52301\uff1f\u79ef\u5206\u533a\u57df\u4e3a
0\u2264\u03b8\u2264\u03c0/2
0\u2264r\u2264cos\u03b8
r=cos\u03b8\u8868\u793a\u5706
x^2+y^2=x
\u6240\u4ee5\uff0cD\u8868\u793a\u4e0a\u534a\u5706\uff0c
\u4e0a\u534a\u5706\u8868\u793a\u6210\u76f4\u89d2\u5750\u6807\uff0c\u5c31\u662f
0\u2264x\u22641
0\u2264y\u2264\u221a(x-x^2)
\u89e3\uff1a
\u222b(x-e^x)dx\uff0c[x\uff1a-1\u21920]
=\u222bxdx-\u222b(e^x)dx\uff0c[x\uff1a-1\u21920]
=(x^2)/2-e^x+C\uff0c[x\uff1a-1\u21920]
=[(0)^2]/2-e^(0)-{[(-1)^2]/2-e^(-1)}
=-1-1/2+1/e
=(1/e)-3/2
∫ x*e^(-x/a) dx
＝ ∫ －a * x * d [ e^(-x/a) ]
＝ －a * x * e^(-x/a) + ∫ a * e^(-x/a) dx
＝－a * x * e^(-x/a) － a^2 * e^(-x/a)
代入后容易得到积分结果＝a^2
(假定a>0,a<0发散）
用r函数=y^2
的法国大范甘迪
绛旓細姝ゅy鐢╝琛ㄧず,鐢ㄥ垎閮ㄧН鍒嗗叕寮忥細鈭 x*e^(-x/a) dx 锛 鈭 锛峚 * x * d [ e^(-x/a) ]锛 锛峚 * x * e^(-x/a) + 鈭 a * e^(-x/a) dx锛濓紞a * x * e^(-x/a) 锛 a^2 * e^(-x/a)浠ｅ叆鍚庡鏄撳緱鍒扮Н鍒嗙粨鏋滐紳a^2(鍋囧畾a>0,a...
绛旓細x*e^(-x)|(0,+鈭)x->+鈭 lim x/e^x=lim1/e^x=0 x=0鍘熷紡=0 鎵浠ヤ袱鑰呭樊涓0
绛旓細鈭玿e^(-k²x²) dx 0鍒版鏃犵┓ = (1/2)鈭玡^(-k²x²) d(x²)= (1/2)(1/-k²)鈭玡^(-k²x²) d(-k²x²)= -1/(2k²)*e^(-k²x²)= -1/(2k²)*(0-e^0)= 1/(2k²)...
绛旓細=-xe^(-x)-e^(-x)=-(x+1)e^(-x)甯﹀叆闆剁偣=0杞寲鎴愬畾绉垎锛屽嵆鎵寰椾唬鏁板紡+1=-(x+1)e^(-x)+1銆傚師绉垎鍚庡崐閮ㄥ垎姹傚師鍑芥暟锛屽墠鍗婇儴鍒嗗嵆x姹鍏跺鏁帮紝鍒╃敤绉垎鍏紡鍗冲彲锛屽彲瀵圭浜岃浠ｆ暟寮忕涓閮ㄥ垎姹傚楠岃瘉銆
绛旓細鈭祔=鈭玿e^(-x)dx=(-x-1)e^(-x)+C 鍙栦竴涓師鍑芥暟F(x)=(-x-1)e^(-x)lim(x鈫+鈭)F(x)=-x/e^x-1/e^x =lim(x鈫+鈭)-1/e^x-0 =0 F(0)=-1 鈭粹埆[0,+鈭瀅xe^(-x)dx=lim(x鈫+鈭)F(x)-F(0)=1
绛旓細s = 0 鏃讹紝 I = 鈭玿dx = x^2/2 + C;s 鈮 0 鏃讹紝 I = 鈭玿e^(-sx)dx = (-1/s)鈭玿de^(-sx)= (-1/s)xe^(-sx) - (-1/s)鈭玡^(-sx)dx = (-1/s)xe^(-sx) - (1/s^2)e^(-sx) + C 鎵姹傚畾绉垎鍙戞暎銆
绛旓細浠0鍒版鏃犵┓澶*x*锛坋鐨勮礋锛坸鐨勫钩鏂癸級锛=鈭(x^2)*e^(-x^2)dx =(鈭玿*e(-x^2)dx^2)/2 =-(鈭玿 d(e^(-x^2)))/2 =-x*e^(-x^2)/2+(鈭玡^(-x^2) dx)/2 =0+(鈭玡^(-x^2) dx)/2 浠=(鈭玡^(-x^2) dx)/2=(鈭玡^(-y^2) dy)/2 t*t=((鈭玡^(-...
绛旓細鑻ュ畾绉垎瀛樺湪锛屽垯瀹冩槸涓涓叿浣撶殑鏁板硷紙鏇茶竟姊舰鐨勯潰绉級锛岃屼笉瀹氱Н鍒嗘槸涓涓嚱鏁拌〃杈惧紡锛屽畠浠粎浠呭湪鏁板涓婃湁涓涓绠楀叧绯伙紙鐗涢】-鑾卞竷灏艰尐鍏紡锛夛紝鍏跺畠涓鐐瑰叧绯婚兘娌℃湁锛佷竴涓嚱鏁帮紝鍙互瀛樺湪涓嶅畾绉垎锛岃屼笉瀛樺湪瀹氱Н鍒嗭紱涔熷彲浠ュ瓨鍦ㄥ畾绉垎锛岃屼笉瀛樺湪涓嶅畾绉垎銆備竴涓繛缁嚱鏁帮紝涓瀹氬瓨鍦ㄥ畾绉垎鍜屼笉瀹氱Н鍒嗭紱鑻ュ彧鏈...
绛旓細鍥炵瓟濡備笅锛氬鏋滀竴涓嚱鏁癴鍙Н锛岄偅涔堝畠涔樹互涓涓父鏁板悗浠嶇劧鍙Н銆傚鏋滃嚱鏁癴鍜実鍙Н锛岄偅涔堝畠浠殑鍜屼笌宸篃鍙Н銆傚鏋滀竴涓嚱鏁癴鍦ㄦ煇涓尯闂翠笂榛庢浖鍙Н锛屽苟涓斿湪姝ゅ尯闂翠笂澶т簬绛変簬闆躲傞偅涔堝畠鍦ㄨ繖涓尯闂翠笂鐨勭Н鍒嗕篃澶т簬绛変簬闆躲傚鏋渇鍕掕礉鏍煎彲绉苟涓斿嚑涔庢绘槸澶т簬绛変簬闆讹紝閭ｄ箞瀹冪殑鍕掕礉鏍肩Н鍒嗕篃澶т簬绛変簬闆躲
绛旓細浣犳槸璇村xe^x 绉垎鍚楋紵杩欎釜鍑芥暟鍦0鍒版鏃犵┓鐨勫尯闂翠笂骞夸箟绉垎鏄笉鏀舵暃鐨勩備篃灏辨槸鏃犵┓澶с傝瘉鏄庡涓嬶細xe^x=x(1+x+鈥︹︼級>x;x鐨勭Н鍒嗕负0.5x^2,涓嶆敹鏁涳紝浠庤屾墍姹傚嚱鏁颁篃涓嶆敹鏁
