∫（π/2,0)(x^e)*(cosx)dx=?
\u222b(\u4e0b0,\u4e0a\u03c0/4) x/(cosx)^2 dx\u222bxdx/(cosx)^2= \u222bxdtanx=xtanx-\u222btanxdx=xtanx-\u222bsinxdx/cosx=xtanx+lncosx+C
\u222b[0, \u03c0/4] xdx/(cosx)^2=\u03c0/4-ln\u221a2
=\u222bx(secx)^2dx
=\u222bxdtanx
=xtanx-\u222btanxdx
=xtanx-\u222bsinx/cosxdx
=xtanx+\u222bdcosx/cosx
=xtanx+ln|cosx|+C
2∫e^xcosxdx=e^xsinx+e^xcosx
∫e^xcosxdx=(1/2)e^x(sinx+cosx)
∫[π/2,0] e^xcosxdx= (1/2)-(1/2)e^(π/2)
绛旓細杩欐槸鐢ㄤ簡瀹氱Н鍒嗗尯闂村啀鐜板叕寮忥紝鍙互涓嶆敼鍙樼Н鍒嗗尯鍩熺殑鎯呭喌涓嬪琚Н鍑芥暟杩涜鏀归狅紝鏂逛究绉垎璁＄畻銆鈭(0,蟺)xsin²xdx =鈭(0,蟺)(蟺-x)sin²(蟺-x)dx =鈭(0,蟺)(蟺-x)sin²(x)dx =鈭(0,蟺)[蟺sin²(x)dx-鈭(0,蟺)xsin²xdx 鈭2鈭(0,蟺)xsin²xdx...
绛旓細鍥犱负鈭(0, 蟺)f(sinx)dx锛濃埆(0, 蟺/2)f(sinx)dx锛鈭(蟺/2, 蟺)f(sinx)dx銆傚浜庣浜屼釜绉垎锛屼护x锛澫-t, 鍒欌埆(蟺/2, 蟺)f(sinx)dx锛濃埆(蟺/2, 0)f(sint)(-dt)锛濃埆(0, 蟺/2)f(sint)dt锛濃埆(0, 蟺/2)f(sinx)dx銆傛墍浠モ埆(0, 蟺)f(sinx)dx锛2鈭(0, 蟺/2)f(sinx...
绛旓細銆愭暟瀛︿箣缇庛戝洟闃熶负鎮ㄨВ绛旓紝鑻ユ湁涓嶆噦璇疯拷闂紝濡傛灉瑙ｅ喅闂璇风偣涓嬮潰鐨勨滈変负婊℃剰绛旀鈥濄
绛旓細杩欐槸涓涓叕寮忥紝鎺ㄥ濡備笅锛
绛旓細鈭xcosxdx =鈭玿dsinx =x*sinx-鈭玸inxdx =xsinx+cosx+C 鎵浠 鈭玔蟺, -蟺] xcosxdx =xsinx+cosx |[蟺, -蟺]=(蟺sin蟺+cos蟺)-[(-蟺)sin(-蟺)+cos(-蟺)]=-1-(-1)=0 瀹為檯涓婏紝鏈夋洿绠鍗曠殑鏂规硶 xcosx 鏄竴涓鍑芥暟 鍥惧儚鍏充簬鍘熺偣瀵圭О锛屽埄鐢ㄥ畾绉垎鐨勫嚑浣曟剰涔夛紝寰 鈭玔蟺, -蟺] ...
绛旓細鍥炵瓟锛-pi 鍒皃i 鍖洪棿,f(x)涓0,绉垎涓洪浂,鎵浠ヤ笉鐢ㄨ绠
绛旓細濡備笅锛歏x = 鈭 蟺[ f(x)]^2 dx 鏄 [a, b] 涓婃洸杈规褰㈢粫 x 杞存棆杞綋鐨勪綋绉叕寮忋俈y = 鈭 2蟺xf(x) dx 鏄 [a, b] 涓婃洸杈规褰㈢粫 y 杞存棆杞綋鐨勪綋绉叕寮忋傚悗鑰呬竴鑸篃鐢 Vy = 鈭 蟺[ g(y)]^2 dy銆備綋绉殑姒傚康 褰撶墿浣撳崰鎹殑绌洪棿鏄笁缁寸┖闂存椂锛屾墍鍗犵┖闂寸殑澶у皬鍙仛璇ョ墿浣撶殑浣撶Н銆
绛旓細鎵浠鈭(蟺/2鈫捪)f(sinx)dx =鈭(蟺/2鈫0)f(sin(蟺-t))d(蟺-t)=-鈭(蟺/2鈫0)f(sint)dt =鈭(0鈫捪/2)f(sint)dt =鈭(0鈫捪/2)f(sinx)dx(瀹氱Н鍒嗕笌绉垎鍙橀噺鏃犲叧锛変簬鏄埆(0鈫捪)f(sinx)dx=鈭(0鈫捪/2)f(sinx)dx+鈭(蟺/2鈫捪)f(sinx)dx=2鈭(0鈫捪/2)f(sint)dt...
绛旓細鍥犱负鏄弽甯哥Н鍒嗭紝鍦▁=蟺/2鐐硅繘琛屽垎娈碉細鈭(蟺/2, 0)tanxdx+鈭(蟺, 蟺/2)tanxdx 鈭(蟺/2, 0)tanxdx =ln|secx|(蟺/2, 0)涓婇檺鏃剁殑鍊硷細lim(x->蟺/2-)ln|secx|=鈭 鎵浠ユ湰瀹氱Н鍒嗗彂鏁ｃ
绛旓細杩欎釜浣撶Н绛変簬2蟺xcosx鍦蓟0锛屜/2锛戒笂鐨勫畾绉垎锛岀瓟妗堟槸2蟺(蟺/2-1)銆=-2蟺鈭(蟺/2鍒0)tdsint =-2蟺[tsint|(蟺/2鍒0)-鈭(蟺/2鍒0)sintdt]=蟺²+鈭(蟺/2鍒0)sintdt =2蟺(蟺/2-1)銆
