根下1-cosx分之一的不定积分 根号下1+cosx怎么积分
\u6839\u53f7\u4e0b1 cosx\u7684\u4e0d\u5b9a\u79ef\u5206\u5177\u4f53\u56de\u7b54\u5982\u56fe\uff1a
\u4e00\u4e2a\u51fd\u6570\uff0c\u53ef\u4ee5\u5b58\u5728\u4e0d\u5b9a\u79ef\u5206\uff0c\u800c\u4e0d\u5b58\u5728\u5b9a\u79ef\u5206\uff0c\u4e5f\u53ef\u4ee5\u5b58\u5728\u5b9a\u79ef\u5206\uff0c\u800c\u6ca1\u6709\u4e0d\u5b9a\u79ef\u5206\u3002\u8fde\u7eed\u51fd\u6570\uff0c\u4e00\u5b9a\u5b58\u5728\u5b9a\u79ef\u5206\u548c\u4e0d\u5b9a\u79ef\u5206\u3002
\u82e5\u5728\u6709\u9650\u533a\u95f4[a,b]\u4e0a\u53ea\u6709\u6709\u9650\u4e2a\u95f4\u65ad\u70b9\u4e14\u51fd\u6570\u6709\u754c\uff0c\u5219\u5b9a\u79ef\u5206\u5b58\u5728\uff1b\u82e5\u6709\u8df3\u8dc3\u3001\u53ef\u53bb\u3001\u65e0\u7a77\u95f4\u65ad\u70b9\uff0c\u5219\u539f\u51fd\u6570\u4e00\u5b9a\u4e0d\u5b58\u5728\uff0c\u5373\u4e0d\u5b9a\u79ef\u5206\u4e00\u5b9a\u4e0d\u5b58\u5728\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u7531\u4e8e\u5728\u4e00\u4e2a\u533a\u95f4\u4e0a\u5bfc\u6570\u6052\u4e3a\u96f6\u7684\u51fd\u6570\u5fc5\u4e3a\u5e38\u6570\uff0c\u6240\u4ee5G(x)-F(x)=C\u2019(C\u2018\u4e3a\u67d0\u4e2a\u5e38\u6570)\u3002
\u8fd9\u8868\u660eG(x)\u4e0eF(x)\u53ea\u5dee\u4e00\u4e2a\u5e38\u6570\uff0c\u56e0\u6b64\uff0c\u5f53C\u4e3a\u4efb\u610f\u5e38\u6570\u65f6\uff0c\u8868\u8fbe\u5f0fF(x)+C\u5c31\u53ef\u4ee5\u8868\u793af(x)\u7684\u4efb\u610f\u4e00\u4e2a\u539f\u51fd\u6570\u3002\u4e5f\u5c31\u662f\u8bf4f(x)\u7684\u5168\u4f53\u539f\u51fd\u6570\u6240\u7ec4\u6210\u7684\u96c6\u5408\u5c31\u662f\u51fd\u6570\u65cf{F(x)+C|-\u221e<C<+\u221e}\u3002
\u7531\u6b64\u53ef\u77e5\uff0c\u5982\u679cF(x)\u662ff(x)\u5728\u533a\u95f4I\u4e0a\u7684\u4e00\u4e2a\u539f\u51fd\u6570\uff0c\u90a3\u4e48F(x)+C\u5c31\u662ff(x)\u7684\u4e0d\u5b9a\u79ef\u5206\uff0c\u5373\u222bf(x)dx=F(x)+C\u3002
\u8bbef(x)\u5728\u533a\u95f4[a\uff0cb]\u4e0a\u8fde\u7eed\uff0c\u5219f(x)\u5728[a\uff0cb]\u4e0a\u53ef\u79ef\u3002\u8bbef(x)\u533a\u95f4[a\uff0cb]\u4e0a\u6709\u754c\uff0c\u4e14\u53ea\u6709\u6709\u9650\u4e2a\u95f4\u65ad\u70b9\uff0c\u5219f(x)\u5728[a\uff0cb]\u4e0a\u53ef\u79ef\u3002\u8bbef(x)\u5728\u533a\u95f4[a\uff0cb]\u4e0a\u5355\u8c03\uff0c\u5219f(x)\u5728[a\uff0cb]\u4e0a\u53ef\u79ef\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1--\u4e0d\u5b9a\u79ef\u5206
\u5177\u4f53\u56de\u7b54\u5982\u4e0b\uff1a
\u222b \u221a\uff081+cosx\uff09dx
= 2\u222b \uff3b\u221a2\uff08cosx/2\uff09^2\uff3dd\uff08x/2\uff09
=2\u221a2\u222b \u2502cosx/2\u2502d\uff08x/2\uff09
=2\u221a2\u2502sinx/2\u2502+C
\u4e0d\u5b9a\u79ef\u5206\u7684\u610f\u4e49\uff1a
\u4e00\u4e2a\u51fd\u6570\uff0c\u53ef\u4ee5\u5b58\u5728\u4e0d\u5b9a\u79ef\u5206\uff0c\u800c\u4e0d\u5b58\u5728\u5b9a\u79ef\u5206\uff0c\u4e5f\u53ef\u4ee5\u5b58\u5728\u5b9a\u79ef\u5206\uff0c\u800c\u6ca1\u6709\u4e0d\u5b9a\u79ef\u5206\u3002\u8fde\u7eed\u51fd\u6570\uff0c\u4e00\u5b9a\u5b58\u5728\u5b9a\u79ef\u5206\u548c\u4e0d\u5b9a\u79ef\u5206\u3002
\u82e5\u5728\u6709\u9650\u533a\u95f4[a,b]\u4e0a\u53ea\u6709\u6709\u9650\u4e2a\u95f4\u65ad\u70b9\u4e14\u51fd\u6570\u6709\u754c\uff0c\u5219\u5b9a\u79ef\u5206\u5b58\u5728\uff1b\u82e5\u6709\u8df3\u8dc3\u3001\u53ef\u53bb\u3001\u65e0\u7a77\u95f4\u65ad\u70b9\uff0c\u5219\u539f\u51fd\u6570\u4e00\u5b9a\u4e0d\u5b58\u5728\uff0c\u5373\u4e0d\u5b9a\u79ef\u5206\u4e00\u5b9a\u4e0d\u5b58\u5728\u3002
如图所示：
绛旓細1/(1+cosx)=0.5^2 鈭玠x/(1+cosx)锛濃埆0.5^2dx 锛濃埆锛籹ec(x/2)]^2d0.5x 锛濃埆dtan(x/2)=tan(x/2)+c
绛旓細3銆佸父鐢ㄤ笉瀹氱Н鍒嗗叕寮忊埆1dx=x+C銆佲埆1/xdx=ln|x|+C銆佲埆cosxdx=sinx+C銆佲埆sinxdx=-cosx+C銆1/鈭歺鐨勪笉瀹氱Н鍒嗘槸浠涔堟牴鍙穢鐨勪笉瀹氱Н鍒嗘槸锛氫笁鍒嗕箣浜屽嶇殑x鐨勪簩鍒嗕箣涓夋鏂广傚叿浣撳涓嬶細鍙互鏄綔涓轰竴涓暣浣,濡=1-x^2鍗虫眰f鐨勮鏄=鏍鐨勮鐢熺墿锛屼负f'=鈥滀箻浠=1/涔樹互-涓槸=1-x^涓や唬灏卞彲浠ヨ繘...
绛旓細缁撴灉鏄 (1/2)[arcsinx + x鈭(1 - x²)] + C x = sin胃锛宒x = cos胃 d胃鈭 鈭(1 - x²) dx = 鈭 鈭(1 - sin²胃)(cos胃 d胃) = 鈭 cos²胃 d胃= 鈭 (1 + cos2胃)/2 d胃 = 胃/2 + (sin2胃)/4 + C= (arcsinx)/2 + (sin胃cos胃...
绛旓細1+cosx=2[cos(x/2)]^2 1/(1+cosx)=0.5[sec(x/2)]^2 鈭玠x/(1+cosx)=鈭0.5[sec(x/2)]^2dx =鈭玔sec(x/2)]^2d0.5x =鈭玠tan(x/2)=tan(x/2)+c
绛旓細鍙互鎶奵osX鍒掍负cosX/2鐨勫钩鏂-sinX/2鐨勫钩鏂癸紝1鍙垝涓篶osX/2鐨勫钩鏂+sinX/2鐨勫钩鏂癸紝鎵浠1-cosX鍙垝涓2sinX/2鐨勫钩鏂癸紝鎵浠ユ牴鍙涓1-cosX灏卞彲鍒掍负鏍瑰彿2涔樹互sinX/2銆涓嶅畾绉鍒嗙殑杩囩▼锛氾紙1+cosx)^2=1+2cosx+cos^2x=1/2cos2x+2cosx+3/2 鏁呭叾鍘熷嚱鏁颁负锛1/4sin2x+2sinx+3/2x+a(甯告暟)...
绛旓細銆愭眰瑙ｈ繃绋嬨戙愭湰棰樼煡璇嗙偣銆1銆佲埆(1/sinx)dx鍏紡鐨勬帹瀵笺2銆佷笉瀹氱Н鍒嗐傝f(x)鍦ㄦ煇鍖洪棿I涓婃湁瀹氫箟锛屽鏋滃瓨鍦ㄥ嚱鏁癋(x)锛屼娇寰楀浜庝换涓x鈭圛锛屾垚绔婩'(x)=f(x)锛屽垯绉癋(x)鏄痜(x)鐨勫師鍑芥暟锛屼笖f(x)鐨勪笉瀹氱Н鍒嗕负 鈭玣(x)dx=F(x)+C 寮忎腑锛氣埆鈥斺绉垎鍙凤紝f(x)dx鈥斺旇绉紡锛宖(x)...
绛旓細绉垎杩囩▼涓 浠 = sin胃锛屽垯dx = cos胃 d胃 鈭垰(1-x²)dx =鈭垰(1-sin²胃)(cos胃 d胃)=鈭玞os²胃d胃 =鈭(1+cos2胃)/2d胃 =胃/2+(sin2胃)/4+C =(arcsinx)/2+(sin胃cos胃)/2 + C =(arcsinx)/2+(x鈭(1 - x²))/2+C =(1/2)[arcsinx...
绛旓細浠osx=t浠ｅ叆鍚庯紝缁忓寲绠锛岃绉嚱鏁板彉涓簍/(1-t)[(1+t)^2]锛岀Н鍒嗗寲鎴愪簡鏈夌悊鍑芥暟鐨勪笉瀹氱Н鍒嗐傚啀璁 t/(1-t)[(1+t)^2]=A/1-t +B/1+t +C/(1+t)^2锛屽彲鍒嗗埆姹傚嚭A銆丅銆丆鐨勫硷紝浠ｅ洖鍚庡嵆鍙眰鍑轰笉瀹氱Н鍒嗙粨鏋滀负1/4* ln|1-cosx|-1/2* ln|1+cosx|-1/2(1+t)+C.娉細鏃...
绛旓細璇︽儏濡傚浘鎵绀 鏈変换浣曠枒鎯戯紝娆㈣繋杩介棶
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