大学高数求不定积分 大学高数不定积分，求完整过程
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\u7b54\u6848\u5982\u56fe\u6240\u793a
\u5f0f\u5b50\u4e24\u8fb9\u5bf9x\u6c42\u5bfc\uff0c\u5219\u6709g(f(x))f'(x)=e^x(x^2+2x) \u6ce8\u610f\u5230g(f(x))=x\uff0c\u6240\u4ee5\u5c31\u6709f'(x)=e^x(x+2) \u4e24\u8fb9\u79ef\u5206\uff0c\u5c31\u53ef\u4ee5\u628af(x)\u89e3\u51fa\u6765\uff0c\u4f46\u662f\u8981\u6ce8\u610ff(0)=0\u8c03\u6574\u4e00\u4e0b\u5e38\u6570\u9879
∫sin²x/cos³xdx=∫(sinxtanx)/cos²xdx
=∫(sinxtanx)d(tanx)
=(1/2)∫sinxd(tan²x)
=(1/2)sinxtan²x+(1/2)∫tan²xd(sinx)
=(1/2)sinxtan²x+(1/2)∫sin²x/cosxdx
=(1/2)sinxtan²x+(1/2)∫(1-cos²x)/cosxdx
=(1/2)sinxtan²x+(1/2)∫(1/cosx-cosx)dx
=(1/2)sinxtan²x-(1/2)sinx+(1/2)∫(1/cosx)dx
=(1/2)sinxtan²x-(1/2)sinx+(1/2)(ln|secx+tanx|)+C
绛旓細(2)澶氭搴旂敤鍒嗛儴绉垎娉曪紝姣忓垎閮ㄧН鍒嗕竴娆″緱浠ョ畝鍖栵紝鐩磋嚦鏈鍚庢眰鍑恒(3)鐢ㄥ垎閮ㄧН鍒嗘硶鏈夋椂鍙鍑衡埆f(x)dx鐨勬柟绋嬶紝鐒跺悗瑙ｅ嚭銆(4)鏈夋椂鐢ㄥ垎閮ㄧН鍒嗘硶鍙鍑洪掓帹鍏紡 鍦澶у楂樻暟瀛︿範涓嶅畾绉垎鐢ㄥ垎閮ㄧН鍒嗘硶鏃讹紝涓鑸儏鍐典笅锛屾帉鎻″墠3绉嶅嵆鍙紝鍗充娇鑰冭瘯鏈鍚庣殑鍘嬭酱棰樼洰涔熼冧笉鍑鸿繖涓寖鍥达紝瀵逛簬鑰冪爺鐨勫瀛(鍙鏁颁竴)...
绛旓細=鈭玸ec²tdt/[(2tan²t+1)sect]=鈭玠t/[cost((2sin²t/cos²t)+1)]=鈭玞ostdt/[((2sin²t+cost²)]=鈭玔1/(1+sin²t)]d(sint)=arctan(sint)+C 涓夎鏇挎崲鏈塻int=x/鈭(1+x²)鎵浠ュ師涓嶅畾绉垎 鈭玠x/(2x^2+1)(x^2+1)^(1/2)=...
绛旓細鐢ㄤ笁瑙掑嚱鏁颁唬鎹㈠拰鍒嗛儴绉垎锛氬師寮=鈭6t^5dt/[t^3*鈭(1+t^2)]=6鈭玹^2dt/鈭(1+t^2)璁総=tan胃锛宒t=(sec胃)^2d胃.鍘熷紡=6鈭(tan胃)^2*(sec胃)^2d胃/sec胃 =6鈭玔(sec胃)^2-1]sec胃d胃 =6鈭(sec胃)^3d胃-6鈭玸ec胃d胃,鈭(sec胃)^3d胃=鈭(sec胃)dtan胃 =sec胃tan胃...
绛旓細1銆佷护x=1/t dx=-dt/t^2 鍘熷紡=-鈭玹dt/鈭(t^4+1)=-1/2*鈭玠(t^2)/鈭歔(t^2)^2+1]=-1/2*ln|t^2+鈭(t^4+1)|+C =-1/2*ln|1/x^2+鈭(1/x^4+1)|+C 2銆佷护x=sint dx=costdt 鍘熷紡=鈭玞ostdt/(sint+cost)浠=鈭玞ostdt/(sint+cost) B=鈭玸intdt/(sint+...
绛旓細鍥炵瓟锛=鈭1/sintdt =鈭玞sctdt =ln|csct-cott|+C =ln|鈭(x^2+1)-1|-ln|x|+C
绛旓細绉垎:sin(lnx)dx (鍒嗛儴绉垎)=xsin(lnx)-绉垎:xcos(lnx)/xdx =xsin(lnx)-绉垎:cos(lnx)dx (鍐嶅垎閮ㄧН鍒)=xsin(lnx)-xcos(lnx)-绉垎:xsin(lnx)/xdx =xsin(lnx)-xcos(lnx)-绉垎:sin(lnx)dx 璁惧師鏉ョ殑绉垎涓篞 鍒欐湁:Q=xsin(lnx)-xcos(lnx)-Q 鎵浠 2Q=xsin(lnx)-xcos(lnx)鎵浠 Q...
绛旓細2.鍒嗛儴绉垎锛氣埆ln(1+x^2)dx =xln(1+x^2)-鈭2x^2/(1+x^2)dx =xln(1+x^2)-[鈭2dx-鈭2/(1+x^2)dx]=xln(1+x^2)-2x+2arctanx+C 3.鈭(xcos2x)dx =(1/2)鈭玿dsin2x =(1/2)xsin2x-(1/2)鈭玸in2xdx =(1/2)xsin2x-(1/4)鈭玸in2xd2x =(1/2)xsin2x+(1/4)...
绛旓細鏇寸畝鍗曠殑鏂规硶灏辨槸璁板叕寮忎簡
绛旓細绛旀濡傚浘鎵绀
绛旓細鏈鍚庝竴姝ワ紝鍒嗗埆姹傜Н鍒锛屽緱1/2x^2+2x+31/8ln(x-1)-9/4/(x-1)-1/4/(x-1)^2+1/8ln(x+1)+c(甯告暟锛夋病楠岀畻锛屽鏋滅粨鏋滀笉鏄彲浠ヨ嚜宸卞啀绠楃畻锛屼絾鏄濊矾涓瀹氭槸瀵圭殑 绗竴棰樻病缁嗙畻浜嗭紝鍒嗘瘝鍒嗚В鍥犲紡鏄痻(x-1)(x-2)(x+1)(x+2),鍚庨潰鍚岀浜岄 鏈夐棶棰樺彲浠ュ啀闂紝鏈涢噰绾硂(鈭鈭)o~
