这道题的不定积分怎么解的？ 这道不定积分题怎么算？
\u8fd9\u9053\u4e0d\u5b9a\u79ef\u5206\u600e\u4e48\u89e3\u4ee4lnx=t\uff0c\u89e3\u4e0d\u5b9a\u79ef\u5206\u7ed3\u679c\u4e3at^2/2,\u518d\u56de\u4ee3lnx\uff0c\u7ed3\u679c\u4e3a\uff08lnx\uff09^2/2
\u4e71\u4e03\u516b\u7cdf\u7684\u7b54\u6848\u771f\u591a\u2026\u2026\u8fc7\u7a0b\u8be6\u60c5\u2026\u2026\u5982\u56fe\u6240\u793a\u2026\u2026
原式 = ∫(u^3-1) / (u+1) * 2udu
=∫(2u^3-2u^2+2u-4+4/(u+1))du
= 1/2*u^4 - 2/3*u^3 + u^2 - 4u + 4ln|u+1|+C
= 1/2*x^2 - 2/3*x^(3/2) + x - 4√x + 4ln(√x+1) + C
绛旓細浠 u = 鈭歺锛屽垯 x = u^2锛宒x = 2udu锛屽師寮 = 鈭(u^3-1) / (u+1) * 2udu =鈭(2u^3-2u^2+2u-4+4/(u+1))du = 1/2*u^4 - 2/3*u^3 + u^2 - 4u + 4ln|u+1|+C = 1/2*x^2 - 2/3*x^(3/2) + x - 4鈭歺 + 4ln(鈭歺+1) + C ...
绛旓細鏂规硶濡備笅锛岃浣滃弬鑰冿細鑻ユ湁甯姪锛岃閲囩撼銆
绛旓細瑙ｉ杩囩▼濡備笅鍥撅細璁颁綔鈭玣(x)dx鎴栬呪埆f锛堥珮绛夊井绉垎涓父鐪佸幓dx锛夛紝鍗斥埆f(x)dx=F(x)+C銆傚叾涓埆鍙仛绉垎鍙凤紝f(x)鍙仛琚Н鍑芥暟锛寈鍙仛绉垎鍙橀噺锛宖(x)dx鍙仛琚Н寮忥紝C鍙仛绉垎甯告暟鎴栫Н鍒嗗父閲忥紝姹傚凡鐭ュ嚱鏁鐨勪笉瀹氱Н鍒嗙殑杩囩▼鍙仛瀵硅繖涓嚱鏁拌繘琛屼笉瀹氱Н鍒嗐
绛旓細鏂规硶涓 1銆佸ぇ澶氭暟澶氶」寮忛傜敤鐨勭Н鍒嗗叕寮忋傛瘮濡傚椤瑰紡锛歽 = a*x^n.銆2銆佺郴鏁伴櫎浠(n+1)锛岀劧鍚庢寚鏁板姞涓1銆傛崲鍙ヨ瘽璇磞 = a*x^n 鐨勭Н鍒嗘槸y = (a/n+1)*x^(n+1).銆3銆佸浜庝笉瀹氱Н鍒嗭紝涓涓椤瑰紡瀵瑰簲澶氫釜锛屾墍浠ヨ鍔犱笂绉垎甯告暟C銆傚洜姝ゆ湰渚嬬殑鏈缁堢粨鏋滄槸y = (a/n+1)*x^(n+1) + C銆 ...
绛旓細鍏堟妸寮忓瓙杞寲涓涓嬶紝鍐嶅埄鐢ㄥ父鐢绉垎鍏紡:
绛旓細瑙ｏ細
绛旓細limx->0锛寋ln(1+x)}/x =limx->0锛(1/x){ln(1+x)} =limx->0锛宭n{(1+x)^(1/x)} =ln{limx->0[(1+x)^(1/x)]} =ln1 =0
绛旓細濡傚浘鍏堢敤鍙橀噺浠ｆ崲x=tanu锛屽寲绠鏀瑰啓鍚庡彲姹傚嚭鍘熷嚱鏁锛屾渶鍚庝唬鍥炲埌x鐨勫嚱鏁板舰寮忋
绛旓細杩欎釜涓嶅畾绉垎鏄笉鑳界畻鍑烘潵鐨勩杩欎釜绉垎涓嶆槸鍒濈瓑鍑芥暟 ,涓嶈兘绉垎銆俥(x2锛変负鍋跺嚱鏁帮紝鍦▁灞炰簬0鍒版棤绌锋椂鍗曞锛屼笖鍙戞暎銆傛墍浠ュ叾涓嶅畾绉垎涓烘棤绌凤紝涓嶈兘绉垎銆傛槸娌℃硶鐢ㄨВ鏋愮殑鏂规硶琛ㄧず鐨勩備竴鑸畾鐞 瀹氱悊1锛氳f(x)鍦ㄥ尯闂碵a,b]涓婅繛缁紝鍒檉(x)鍦╗a,b]涓婂彲绉傚畾鐞2锛氳f(x)鍖洪棿[a,b]涓婃湁鐣岋紝涓...
绛旓細瑙:杩欎釜寰楀叿浣撴儏鍐靛叿浣撳垎鏋愶紝璇锋妸鍏蜂綋鐨勪笉瀹氱Н鍒鍏紡棰樼洰鍙戣繃鏉ワ紝鎴戠湅鐪嬨傛渶濂芥槸鍥剧墖锛岃繖鏍锋瘮杈冪洿瑙傛柟渚胯绠椼備緥濡:涓嬪浘 瑙ｅ父寰垎鏂圭▼ 瑙ｅ父寰垎鏂圭▼ 璇峰弬鑰冿紝甯屾湜瀵逛綘鏈夊府鍔╋紒
