求∫(x+sinx)/(1+cosx)dx从0到蟺/2的积分. 那个字是派
∫(x+sinx)/(1+cosx)dx=∫ (x+2sin x/2cos x/2)/(2cos^2 x/2) dx=1/2 ∫ xsec^2 x/2 dx+ ∫ tan x/2 dx= ∫ x d tan x/2 + ∫ tan x/2 dx= xtanx/2- ∫ tan x/2 dx + ∫ tan x/2 dx=xtanx/2 =π/2tanπ/4 -0=π/2绛旓細xsinx绉垎鏄-xcosx+sinx+C銆傝В鏋愶細xsinx 鈭udv=uv-鈭玽du 鈭 xsinx dx = - 鈭 x d(cosx)=-xcosx+鈭 cosx dx =-xcosx+sinx+C 绉垎鎬ц川锛1銆佺Н鍒嗘槸绾挎х殑銆傚鏋滀竴涓嚱鏁癴鍙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2銆佸鏋滀竴涓嚱鏁癴鍦ㄦ煇涓尯闂...
绛旓細=lim[cos(x+螖x/2)sin(螖x/2]/(螖x/2)螖x鈫0 =cosx 脳 1 =cosx 姹俿in x涓巆os x鐨 n 闃跺鏁帮細(sinx)'=cosx (sinx)''=(cosx)'=-sinx=sin(x+pi/2)(sinx)'''=(-cosx)'=sinx=sin(x+3pi/2)(sinx)^(4)=(sinx)'=cosx=sin(x+4pi/2)鈥︹︹︾粡杩囧綊绾冲緱鍒 (sinx)^(n)...
绛旓細鍏蜂綋鍥炵瓟濡備笅锛鈭玿sinxdx =-鈭玿d(cosx)=-xcosx+鈭玞osxdx 锛堝簲鐢ㄥ垎閮ㄧН鍒嗘硶锛=-xcosx+sinx+C 锛圕鏄Н鍒嗗父鏁帮級鍒嗛儴绉垎娉曟槸寰Н鍒嗗涓殑涓绫婚噸瑕佺殑銆佸熀鏈殑璁＄畻绉垎鐨勬柟娉曘傚畠鏄敱寰垎鐨勪箻娉曟硶鍒欏拰寰Н鍒嗗熀鏈畾鐞嗘帹瀵艰岀殑銆傚畠鐨勪富瑕佸師鐞嗘槸灏嗕笉鏄撶洿鎺ユ眰缁撴灉鐨勭Н鍒嗗舰寮忥紝杞寲涓虹瓑浠风殑鏄撴眰鍑虹粨鏋滅殑绉...
绛旓細绉垎濡備笅锛xsinx鐨勭Н鍒嗙瓑浜庯紞xcosx+sinx+C銆傗埉xsinxdx =-鈭畑d(cosx)=-xcosx+鈭甤osxdx =-xcosx+sinx+C銆傜Н鍒嗗叕寮 绉垎鏄井鍒嗙殑閫嗚繍绠楋紝鍗崇煡閬撲簡鍑芥暟鐨勫鍑芥暟锛屽弽姹傚師鍑芥暟銆傚湪搴旂敤涓婏紝绉垎浣滅敤涓嶄粎濡傛锛屽畠琚ぇ閲忓簲鐢ㄤ簬姹傚拰锛岄氫織鐨勮鏄眰鏇茶竟涓夎褰㈢殑闈㈢Н锛岃繖宸у鐨勬眰瑙ｆ柟娉曟槸绉垎鐗规畩鐨勬ц川鍐冲畾...
绛旓細鈭 xsinx dx = - 鈭 x d(cosx)=-xcosx+鈭 cosx dx =-xcosx+sinx+C 涓嶅畾绉垎锛氫笉瀹氱Н鍒嗙殑绉垎鍏紡涓昏鏈夊涓嬪嚑绫伙細鍚玜x+b鐨勭Н鍒嗐佸惈鈭氾紙a+bx锛夌殑绉垎銆佸惈鏈墄^2卤伪锛2鐨勭Н鍒嗐佸惈鏈塧x^2+b锛坅>0锛夌殑绉垎銆佸惈鏈夆垰锛坅²+x^2锛夛紙a>0锛夌殑绉垎銆佸惈鏈夆垰锛坅^2-x^2锛夛紙a>0锛...
绛旓細绗竴閮ㄥ垎鏄痵inx锛岃繖鏄竴涓凡鐭ュ嚱鏁帮紝鍏朵笉瀹氱Н鍒嗗凡缁忕煡閬擄紝鍗硈inx+C1銆傜浜岄儴鍒嗘槸x锛岃繖鏄竴涓竴娆″嚱鏁帮紝鍏朵笉瀹氱Н鍒嗘槸1/2*x^2+C2銆傚洜姝わ紝f锛坸锛=xsinx鐨勪笉瀹氱Н鍒嗘槸锛鈭紙xsinx锛dx=锛坰inx+1/2*x^2锛+C銆傚叾涓瑿鏄Н鍒嗗父鏁般傛荤粨锛氶氳繃浠ヤ笂鎺ㄥ锛屾垜浠緱鍑篺锛坸锛=xsinx鐨勪笉瀹氱Н鍒嗘槸锛坰inx+...
绛旓細鈭玿sinxdx =-鈭玿dcosx =-(xcosx-鈭玞osxdx =-(xcosx-鈭玠sinx)=sinx-xcosx+C
绛旓細=-xcosx+sinx+C 鎵浠sinx鐨勫師鍑芥暟鏄痜(x)=-xcosx+sinx+C (C鏄换鎰忓父鏁)銆傚師鍑芥暟鐨勫畾鐞嗭細鍘熷嚱鏁扮殑瀹氱悊鏄嚱鏁癴(x)鍦ㄦ煇鍖洪棿涓婅繛缁殑璇濓紝閭ｄ箞f(x)鍦ㄨ繖涓尯闂撮噷蹇呬細瀛樺湪鍘熷嚱鏁般傝繖鏄睘浜庡厖鍒嗕笉蹇呰鏉′欢锛岃繕琚彨鍋氭槸鍘熷嚱鏁板瓨鍦ㄥ畾鐞嗭紝瑕佹槸鍑芥暟鏈夊師鍑芥暟鐨勮瘽锛岄偅瀹冪殑鍘熷嚱鏁颁负鏃犵┓澶氫釜銆備妇涓緥瀛愶紝...
绛旓細鈭玿sinxdx =-鈭玿d(cosx)=-xcosx+鈭玞osxdx (搴旂敤鍒嗛儴绉垎娉)=-xcosx+sinx+C (C鏄Н鍒嗗父鏁)銆傚垎閮ㄧН鍒嗘硶鏄井绉垎瀛︿腑鐨勪竴绫婚噸瑕佺殑銆佸熀鏈殑璁＄畻绉垎鐨勬柟娉曘傚畠鏄敱寰垎鐨勪箻娉曟硶鍒欏拰寰Н鍒嗗熀鏈畾鐞嗘帹瀵艰屾潵鐨勩傚畠鐨勪富瑕佸師鐞嗘槸灏嗕笉鏄撶洿鎺ユ眰缁撴灉鐨勭Н鍒嗗舰寮忥紝杞寲涓虹瓑浠风殑鏄撴眰鍑虹粨鏋滅殑绉垎褰㈠紡鐨勩
绛旓細鈭玿sinx dx 鍒╃敤鍒嗛儴绉垎娉 =-鈭玿dcosx =-xcosx+鈭玞osx dx =-xcosx+sinx+c
