∫arccosx^1/2dx不定积分
令x=cos²t,则dx=2cost*(-sint)=-2costsint=-sin2t,t=arccos(x^1/2)∫arccosx^1/2dx
=∫t*(-sin2t)dt
=1/2∫td(cos2t)
=1/2t*cos2t-1/2∫-2sin2tdt
=1/2t*cos2t-1/2cos2t+C
=1/2cos2t(t-1)+C
=1/2(2cos²t-1)(t-1)+C
=1/2(2x-1)[arccos(x^1/2)-1]+C
=(x-1/2)[arccos(x^1/2)-1]+C
=xarccos(x^1/2)-1/2arccos(x^1/2)-x+1/2+C
=xarccos(x^1/2)-1/2arccos(x^1/2)-x+D
绛旓細鈭玜rccosx^1/2dx =鈭玹*锛-sin2t锛塪t =1/2鈭玹d锛坈os2t锛=1/2t*cos2t-1/2鈭-2sin2tdt =1/2t*cos2t-1/2cos2t+C =1/2cos2t锛坱-1锛+C =1/2锛2cos²t-1锛夛紙t-1锛+C =1/2锛2x-1锛塠arccos锛坸^1/2锛-1]+C =锛坸-1/2锛塠arccos锛坸^1/2锛-1]+C =xarccos锛坸...
绛旓細arccos1=0 arccos(-1)=蟺 arccos琛ㄧず鐨勬槸鍙嶄笁瑙掑嚱鏁颁腑鐨勫弽浣欏鸡銆備竴鑸敤浜庤〃绀哄綋瑙掑害涓洪潪鐗规畩瑙掓椂銆傜敱浜庢槸澶氬煎嚱鏁帮紝寰寰鍙栧畠鐨勫崟鍊硷紝鍊煎煙涓篬0锛屜]锛岃浣測=arccosx锛屾垜浠О瀹冨彨鍋氬弽涓夎鍑芥暟涓殑鍙嶄綑寮﹀嚱鏁扮殑涓诲笺傚氨鏄凡鐭ヤ綑寮︽暟鍊硷紝鍙嶆眰瑙掑害锛屽cos(a) = b锛屽垯arccos(b) = a锛涘畠鐨勫兼槸浠...
绛旓細y=arccosx,鈭玜rccosxdx=鈭珁dcosy=ycosy-鈭玞osydy =ycosy-siny+C =xarccosx-鈭(1-x^2)+C
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绛旓細杩欐牱瀛
绛旓細arccosx鐨勫鏁版槸锛-1/鈭(1-x²)銆俛rccos琛ㄧず鐨勬槸鍙嶄笁瑙掑嚱鏁颁腑鐨勫弽浣欏鸡銆備竴鑸敤浜庤〃绀哄綋瑙掑害涓洪潪鐗规畩瑙掓椂銆傛眰瀵兼暟鏃舵寜澶嶅悎娆″簭鐢辨瘮杈冨灞傝捣锛屽悜鍐涓灞備竴灞傚湴瀵逛腑闂村彉閲忔眰瀵兼暟锛岀洿鍒板鑷彉閲忔眰瀵兼暟涓烘銆俛rccosx瀵兼暟璁$畻鏂规硶濡備笅锛氬叾浠栧鏁板叕寮 1銆亂=c(c涓哄父鏁) y'=0 2銆亂=x^n y'=nx^...
绛旓細arccos琛ㄧず鐨勬槸鍙嶄笁瑙掑嚱鏁颁腑鐨勫弽浣欏鸡銆備竴鑸敤浜庤〃绀哄綋瑙掑害涓洪潪鐗规畩瑙掓椂銆傜敱浜庢槸澶氬煎嚱鏁帮紝寰寰鍙栧畠鐨勫崟鍊硷紝鍊煎煙涓篬0锛屜]锛岃浣測=arccosx锛屾垜浠О瀹冨彨鍋氬弽涓夎鍑芥暟涓殑鍙嶄綑寮﹀嚱鏁扮殑涓诲笺傚弽姝e壊鍑芥暟 鍙嶆鍓插嚱鏁版槸鏁拌锛屽睘浜庡弽涓夎鍑芥暟鐨勪竴绉嶃傛寚姝e壊鍑芥暟y=secx鍦ㄥ尯闂碵0锛屜/2)鈭(蟺/2锛屜]涓婄殑...
绛旓細鈭(0->1) arcsinx.arccosx dx =鈭(0->1) arcsinx.[ 蟺/2- arcsinx ] dx =(蟺/2)鈭(0->1) arcsinx dx -鈭(0->1) (arcsinx)^2 dx =(蟺/2) .(蟺/2 - 1) -[(蟺/2)^2 -2]=2-(蟺/2)--- 鈭(0->1) arcsinx dx =[x.arcsinx]|(0->1) -鈭(0->1) x/...
绛旓細鍙嶄綑寮﹀嚱鏁扮殑姹傚 (arccosx)=-1/鈭(1-x^2)鍙嶆鍒囧嚱鏁扮殑姹傚 (arctanx)=1/(1+x^2)鍙嶄綑鍒囧嚱鏁扮殑姹傚 (arccotx)=-1/(1+x^2)涓洪檺鍒跺弽涓夎鍑芥暟涓哄崟鍊煎嚱鏁帮紝灏嗗弽姝e鸡鍑芥暟鐨勫紋闄愬湪-蟺/2鈮鈮は/2锛屽皢y浣滀负鍙嶆寮﹀嚱鏁扮殑涓诲硷紝璁颁负y=arcsinx銆傜浉搴斿湴銆傚弽浣欏鸡鍑芥暟y=arccosx鐨勪富鍊奸檺鍦0...
绛旓細arccosx鐨勫畾涔夊煙x鈭圼-1锛1]銆俛rccos琛ㄧず鐨勬槸鍙嶄笁瑙掑嚱鏁颁腑鐨勫弽浣欏鸡銆傛眰arccosx鐨勫鏁帮紝鍏堟崲涓厓锛屼护t=arccosx锛屾湁cost=x锛屾帴鐫绠楀嚭瀵兼暟涓-1/鏍瑰彿1-x鐨勫钩鏂癸紝鐒跺悗绉垎0鍒皒锛屾渶鍚庡埄鐢紙1+x锛夌殑m娆℃柟灞曞紑銆俛rccosx鐢变簬鏄鍊煎嚱鏁帮紝寰寰鍙栧畠鐨勫崟鍊硷紝鍊煎煙涓篬0锛屜]銆傚弽浣欏鸡鏄竴绉嶅弽涓夎鍑芥暟锛...