用定积分定义计算∫（0,1）sinxdx 利用定积分的性质证明下列不等式 1<∫[π/2 0] sin...
exsinxdx\u7684\u5b9a\u79ef\u5206\u8ba1\u7b97\u673a\u57fa\u7840\u7b54\u6848
\u5982\u56fe
f(x)=sinx/x f'(x)=\uff08xcosx-sinx\uff09/x²=cosx(x-tanx)/x²<0
0\u2264x\u2264 \u03c0/2 \u65f6 sin(\u03c0/2)/(\u03c0/2)\u2264sinx/x0+) sinx/x=1
\u6240\u4ee5\u79ef\u5206 \u222b[\u03c0/2 0] sinxdx/x <\u222b[\u03c0/2 0]dx= \u03c0/2
\u222b[\u03c0/2 0] sinxdx/x > \u222b[\u03c0/2 0] 1/(\u03c0/2)dx=1
您好，答案如图所示：
绛旓細璋冨拰绾ф暟 鈭<n=1,鈭>1/n = 1+1/2+1/3+ ... + 1/n +... 瓒嬩簬鏃犵┓澶э紝 涓嶆敹鏁
绛旓細(x^2+cx+c)^2=x4+c2x2+c2+2cx3+2c2x+2cx2 鍘熷嚱鏁=x5/5+c2x3/3+c2x+cx4/2+c2x2+2cx3/3 銆0锛1銆S=7c2/3+7c/6+1/5 c=-1/4鏃讹紝S鏈灏
绛旓細灏嗗尯闂 [0, 1] 绮剧粏鍒掑垎鎴愭棤鏁颁釜绛夐棿璺濈殑瀛愬尯闂达紝姣忎釜瀛愬尯闂村搴︿负 螖x = 1/n銆傚姣忎釜瀛愬尯闂达紝浠ュ乏绔偣 x_i 涓哄簳杈癸紝鍑芥暟鍊 f(x_i) = x_i 涓洪珮锛屾瀯閫犳垚涓涓繎浼肩殑姊舰銆璁＄畻姣忎釜姊舰鐨勯潰绉 S_i锛岀劧鍚庣疮鍔狅紝寰楀埌绉垎鐨勮繎浼煎 I 鈮 危S_i銆傞殢鐫瀛愬尯闂存暟 n 瓒嬭繎鏃犵┓澶э紝瀹氱Н鍒鐨...
绛旓細杩欓噷杩愮敤鍒颁竴涓叕寮忥細鈭玿^ndx=1/(n+1)x^(n+1)+C 鎵浠ワ細鈭垰xdx锛濃埆x锛撅紙1/2锛塪x锛2/3x^(3/2)锛婥 鈭玿^2dx锛漻^3/3锛婥
绛旓細璁＄畻瀹氱Н鍒甯哥敤鐨勬柟娉曪細鎹㈠厓娉 锛1锛夛紙2锛墄=蠄(t)鍦╗伪,尾]涓婂崟鍊笺佸彲瀵 锛3锛夊綋伪鈮鈮の叉椂锛宎鈮は(t)鈮,涓斚堬紙伪锛=a,蠄(尾)=b 鍒 2.鍒嗛儴绉垎娉 璁緐=u(x),v=v锛坸)鍧囧湪鍖洪棿[a,b]涓婂彲瀵硷紝涓攗鈥诧紝v鈥测垐R锛圼a,b]),鍒欐湁鍒嗛儴绉垎鍏紡锛...
绛旓細缁撴灉涓猴細-1 瑙ｉ杩囩▼濡備笅锛氬師寮=x*lnx-鈭紙1/x)*xdx =xlnx-x+lnx dx =鈭 [0,1] lnx dx =xlnx [0,1]-鈭 [0,1] x*(1/x) dx =0-鈭 [0,1] 1 dx =-1
绛旓細S=鈭玔0,1] cos(1+lnx)dx =xcos(1+lnx)[0,1]-鈭玔0,1] xdcos(1+lnx)=cos1+鈭玔0,1] sin(1+lnx)dx =cos1+xsin(1+lnx)[0,1] -鈭玔0,1] xdsin(1+lnx)=cos1+sin1-鈭玔0,1] cos(1+lnx)dx =cos1+sin1-S S=鈭玔0,1] cos(1+lnx)dx=1/2(cos1+sin1)...
绛旓細鈭玔0,1](x^2-4x+3)dx-鈭玔1,2](x^2-4x+3)dx]=2[(x^3/3-2x^2+3x)[0,1]-(x^3/3-2x^2+3x)[1,2]=2[4/3+2/3]=4 y=x^3 y^2=x 鑱旂珛寰楋細浜ょ偣锛0锛0锛锛岋紙1锛1锛塖=鈭玔0,1](鈭歺-x^3)dx=[2/3 x^(3/2)-x^4/4][0,1]=2/3-1/4=5/12 ...
绛旓細鎵浠/n鐨勫彉鍖栬寖鍥村簲璇ユ槸1/n->1銆傚洜n瓒嬩簬鏃犵┓锛屾晠1/n瓒嬩簬0锛屼簬鏄彲鐭ョН鍒嗗尯闂存槸[0,1],鏋侀檺鍙互鍖栦负sin蟺x鍦╗0,1]涓婄殑瀹氱Н鍒銆傦紙浣犲彲浠ュ掕繃鏉ュ寲涓涓嬭瘯璇曪紝灏嗗尯闂寸瓑鍒嗭紝鐢ㄥ畾涔灏唖in蟺x琛ㄧず鎴愬拰寮忔瀬闄愶紝閭ｄ釜鍏媍鍙栧皬鍖洪棿鐨勭鐐癸級銆傛敞锛氬鏋滃皢k蟺/n浣滀负x锛岄偅涔堣绉嚱鏁板氨鏄痵inx锛岀Н鍒嗗尯闂村彉...
绛旓細浣犲仛鐨勬病鏈夐敊鍟婏紒渚涘弬鑰冿紝璇风瑧绾炽
