利用分部积分法，求不定积分。 利用分部积分法求下列不定积分
\u5229\u7528\u4e0d\u5b9a\u79ef\u5206\u7684\u5206\u90e8\u79ef\u5206\u6cd5\u6c42\u4e0d\u5b9a\u79ef\u5206\u222bxsinx/cos³xdx
\u56e0\u4e3a\uff1a(1/cosx)'=(sinx/cos²x)
\u539f\u5f0f=\u222bx/cosxd(1/cosx) \u5206\u90e8\u79ef\u5206
=x/cos²x-\u222b1/cosxd(x/cosx)
=x/cos²x-\u222b1/cosx*(cosx+xsinx/cos²x)dx
=x/cos²x-\u222b1/cos²xdx-\u222bxsinx/cos³xdx
\u4ee4\u222bxsinx/cos³xdx=F
\u5219F=x/cos²x-\u222b1/cos²xdx-F
2F=x/cos²x-\u222b1/cos²xdx=x/cos²x-\u222bsec²xdx
=x/cos²x-tanx+C
\u6545\u539f\u79ef\u5206=(x/cos²x-tanx)/2+C
=2∫u-2+2/(u+1)du
=u²-4u+4ln(u+1)+C，代入u=√(x+1)
绛旓細=xlnx-x+C
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绛旓細鍒嗘绉垎娉 鍘熷紡=xarctan鈭歺-鈭玿darctan鈭歺 =xarctan鈭歺-鈭玿/(1+x)dx =xarctan鈭歺-鈭(x+1-1)/(1+x)dx =xarctan鈭歺-鈭玔1-1/(1+x)]dx =xarctan鈭歺-x+ln(1+x)+C
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=鈭(u-1)/(u+1)d(u²-1)=2鈭玼-2+2/(u+1)du =u²-4u+4ln(u+1)+C锛屼唬鍏=鈭(x+1)
绛旓細銆
绛旓細瀹氱Н鍒嗗唴 涓涓嶅畾绉垎鐨鍒嗛儴绉垎娉涓鏍凤紝鍙緱鈭玝/a u(x)v'锛坸锛塪x=[鈭玼(x)v'锛坸锛塪x]b/a =[u(x)v(x) - 鈭玽(x)u'(x)dx]b/a =[u(x)v(x)]b/a- 鈭玝/a v(x)u'(x)dx 绠璁颁綔 鈭玝/a uv'dx=[uv]b/a-鈭玝/a u'vdx 鎴栤埆b/a udv=[uv]b/a-鈭玝/a vdu 渚...
绛旓細鍒嗛儴绉垎娉锛庤u=u(x),v=v(x)鏈夎繛缁殑瀵兼暟锛岀敱(uv)'=u'v+uv'锛屽緱uv'=(uv)'-u'v涓よ竟绉垎锛屽悜宸﹁浆|鍚戝彸杞 寮忊憼绉颁负鍒嗛儴绉垎鍏紡锛屼娇鐢ㄥ垎閮ㄧН鍒嗗叕寮姹備笉瀹氱Н鍒鐨勬柟娉曠О涓哄垎閮ㄧН鍒嗘硶锛鍒╃敤鍒嗛儴绉垎鍏紡瑙ｉ鐨勫叧閿槸濡備綍鎭板綋鐨勯夊彇锛岄夊彇鍘熷垯鏄細锛1锛夎瀹规槗姹傚嚭锛庯紙2锛夎姣斿師绉垎鏄撴眰寰楋紟
绛旓細浠わ細F = (-鈭,+鈭)鈭玡^(-x²)dx 鍚屾牱 F= (-鈭,+鈭)鈭玡^(-y²)dy 鐢变簬x,y鏄簰涓嶇浉鍏崇殑鐨绉垎鍙橀噺,鍥犳锛欶² = (-鈭,+鈭)鈭玡^(-x²)dx * (-鈭,+鈭)鈭玡^(-y²)dy = [D]鈭埆e^(-x²)*dx * e^(-y²)*dy = [D]鈭埆e...
绛旓細涓嶅畾绉垎鍒嗛儴绉垎娉浠嬬粛锛氫笉瀹氱Н鍒嗗垎閮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紡鐨勩傚父鐢ㄧ殑鍒嗛儴绉垎鐨勬牴鎹粍鎴愯绉嚱鏁扮殑鍩烘湰鍑芥暟绫诲瀷锛屽皢鍒嗛儴绉垎鐨勯『搴忔暣鐞...
