几道求不定积分的题 求不定积分几道题
\u4e00\u9053\u6c42\u4e0d\u5b9a\u79ef\u5206\u7684\u9898\u76ee\u80fd\u591f\u6709\u591a\u4e2a\u7b54\u6848\u5417\u697c\u4e3b\u8bf4\u5bf9\u4e86\u4e00\u534a\uff1a
1\u3001\u4e0d\u5b9a\u79ef\u5206\u7684\u7ed3\u679c\uff0c\u786e\u5b9e\u4f1a\u6709\u4e0d\u540c\u7684\u7b54\u6848\u3002\u4f46\u662f\uff0c\u6240\u6709\u7684\u7b54\u6848\u90fd\u53ef\u4ee5
\u4e92\u5316\u3002\u4e0b\u9762\u7684\u7b2c\u4e00\u5f20\u56fe\u7247\uff0c\u7ed9\u4e88\u5177\u4f53\u793a\u4f8b\u3002
\u7b2c\u4e8c\u5f20\u56fe\u7247\u662f\u540c\u4e00\u4e2a\u4e0d\u5b9a\u79ef\u5206\u7684\u79ef\u5206\u7ed3\u679c\uff0c\u5b83\u4eec\u6c42\u5bfc\u540e\u662f\u76f8\u540c\u7684\u3002
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2\u3001\u672c\u697c\u4e3b\u4e3e\u7684\u4f8b\u5b50\uff0c\u79ef\u5206\u7684\u7ed3\u679c\u662f\u53cd\u6b63\u5207\u51fd\u6570\uff0c\u800c\u7edd\u4e0d\u53ef\u80fd\u662f\u81ea\u7136\u5bf9\u6570
\u51fd\u6570\u3002\u697c\u4e3b\u5e94\u8be5\u662f\u5bf9 \u222bdx/\uff08ax + b\uff09\u7684\u79ef\u5206\u516c\u5f0f\u7406\u89e3\u9519\u4e86\u3002
.
3\u3001\u82e5\u6709\u7591\u95ee\uff0c\u6b22\u8fce\u8ffd\u95ee\uff0c\u6709\u95ee\u5fc5\u7b54\uff0c\u6709\u7591\u5fc5\u91ca\u3002
\u82e5\u70b9\u51fb\u653e\u5927\uff0c\u56fe\u7247\u5c06\u4f1a\u66f4\u52a0\u6e05\u6670\u3002
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\u3010\u656c\u8bf7\u3011\u656c\u8bf7\u6709\u63a8\u9009\u8ba4\u8bc1\u300a\u4e13\u4e1a\u89e3\u7b54\u300b\u6743\u9650\u7684\u8fbe\u4eba\uff0c\u5343\u4e07\u4e0d\u8981\u5c06\u672c\u4eba\u5bf9\u8be5\u9898\u7684\u89e3\u7b54\u8ba4\u8bc1\u4e3a\u300a\u4e13\u4e1a\u89e3\u7b54\u300b\u3002.\u4e00\u65e6\u88ab\u8ba4\u8bc1\u4e3a\u300a\u4e13\u4e1a\u89e3\u7b54\u300b\uff0c\u6240\u6709\u7f51\u53cb\u90fd\u65e0\u6cd5\u8fdb\u884c\u8bc4\u8bba\u3001\u516c\u8bae\u3001\u7ea0\u9519\u3002\u672c\u4eba\u975e\u5e38\u9700\u8981\u503e\u542c\u5bf9\u6211\u89e3\u7b54\u7684\u5404\u79cd\u53cd\u9988\uff0c\u5373\u4f7f\u662f\u8a00\u8f9e\u6fc0\u70c8\u7684\u3001\u6279\u8bc4\u7684\u3001\u53cd\u9a73\u7684\u8bc4\u8bba\uff0c\u4e5f\u662f\u9700\u8981\u503e\u542c\u7684\u3002
.
\u8bf7\u4f53\u8c05\uff0c\u656c\u8bf7\u5207\u52ff\u8ba4\u8bc1\u3002\u8c22\u8c22\u4f53\u8c05\uff01\u8c22\u8c22\u7406\u89e3\uff01\u8c22\u8c22\uff01\u8c22\u8c22\uff01
\u222barccosxdx=xarccosx-\u222b-x/\u221a(1-x^2) dx\uff08\u5206\u90e8\u79ef\u5206\u6cd5\uff09
=xarccosx-1/2\u222bd(1-x^2)/\u221a(1-x^2)
=xarccosx-\u221a(1-x^2)+C
\u222bsin(lnx)dx=xsin(lnx)-\u222bxcos(lnx)*1/x dx\uff08\u5206\u90e8\u79ef\u5206\u6cd5\uff09
=xsin(lnx)-\u222bcos(lnx)dx
=xsin(lnx)-xcos(lnx)+\u222bx[-sin(lnx)]*1/x dx\uff08\u5206\u90e8\u79ef\u5206\u6cd5\uff09
=xsin(lnx)-xcos(lnx)-\u222bsin(lnx)dx
\u6240\u4ee5\u222bsin(lnx)=x/2[sin(lnx)-cos(lnx)]+C
\u222bx^2arctanxdx=1/3x^3arctanx-1/3\u222bx^3/(x^2+1) dx\uff08\u5206\u90e8\u79ef\u5206\u6cd5\uff09
=1/3x^3arctanx-1/3\u222b[x-x/(x^2+1)]dx
=1/3x^3arctanx-1/6x^2+1/6ln(x^2+1)+C
\u222bln(x+\u221a(1+x^2))dx=xln(x+\u221a(1+x^2))-\u222bx(1+2x/(2\u221a(1+x^2)))/(x+\u221a(1+x^2)) dx\uff08\u5206\u90e8\u79ef\u5206\u6cd5\uff09
=xln(x+\u221a(1+x^2))-\u222bx/\u221a(1+x^2) dx
=xln(x+\u221a(1+x^2))-\u221a(1+x^2)+C
\u222blnx/x dx=\u222blnxd(lnx)=1/2(lnx)^2+C
\u4e2d\u95f4\u7684\u90a3\u4e00\u4e2a\u6ca1\u6709\u770b\u61c2\uff0c\u8bf7\u8ffd\u95ee
3. 令u = secx, du =secx tanx dx
∫ tan^3x secx dx = ∫ (u^2-1) du = u^3/3 - u + C = ......
2. 令u =√x, x=u^2, dx =2u du
∫ arctan√x dx / [√x(1+x)] = ∫ arctanu * 2u du / [ u * (1+u^2)]
= ∫ arctanu du / (1+u^2) = (1/2) (arctanu)^2 + C = (1/2) arctan√x )^2 + C
1. 令u = sinx - cosx, du= (cosx+sinx) dx
∫(sinx+cosx) dx / 3^√(sinx-cosx) = ∫ du / 3^(√u) 令v=√u
= 2 ∫ v * 3^(-v) dv = (-2 /ln3) ∫ v d 3^(-v)
= (-2 /ln3) [ v * 3^(-v) - ∫ 3^(-v) dv ]
= (-2 /ln3) [ v * 3^(-v) + (1/ln3) * 3^(-v) ] + C
=......
的确是很浅的题目啊 ^ ^
(1)∫sinxdx
(2)∫cosxdx
(3)∫x²dx
(4)∫(x³-2x²+1)dx
题呢?
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绛旓細绗竴棰=鈭(X^2+X^(-2)/X^3DX=鈭玐^(-1)+X^(-5)DX=ln|x|-4/x^(-5)+c 绗簩棰樿arcsinx=t锛寈=sint 鈭(arcsinx)^2dx=鈭玹^2costdt=t^2sint-鈭2tsintdt+c=t^2sint+2tcost-鈭2costdt+c=t^2sint+2tcost-2sint+c 绗笁棰樹袱杈规眰瀵煎緱sinx/f(x)=-sinx/(1+cos²x)...
绛旓細1銆侊紳-1/3鈭(2-3x)^3/2 d(-3x)=-1/3*2/5*(2-3x)^(5/2)+C=-2/15*(2-3x)^(5/2)+C 2銆=鈭玡^(3xlna) dx=1/(3lna)*鈭玡^(3xlna) d(3xlna)=1/(3lna)*e^(3xlna)+C=1/(3lna)*a^(3x)+C 3銆=1/2鈭垰(4-x^2) d(x^2)=-1/2鈭(4-x^2)^(1/2) ...
绛旓細1)鈭玡^(-x^2) x^3 dx u = x^2, du = 2 x dx:= 1/2 鈭玡^(-u) u du 鈭玣 dg = f g- 鈭玤 df f = u, dg = e^(-u) du,df = du, g = -e^(-u):= 1/2 鈭玡^(-u) du-(e^(-u) u)/2 鈭玡^(-u) is -e^(-u):= -1/2 e^(-u) u-e...
绛旓細绗簩棰橈細鍏堜袱杈规眰瀵兼暟锛岀劧鍚庨厤鍚圼lnƒ(x)]' = ƒ'(x)/ƒ(x)鐨勫舰寮忥紝鍐嶄袱杈姹傜Н鍒鎵惧嚭ƒ(x)灏唀^(- x)鏀捐繘d閲屽噾鎴愨埆 ƒ(u) du鐨勫舰寮忥紝鐒跺悗灏嗕箣鍓嶆壘鍒扮殑ƒ(x)鐓ф牱浠ｅ叆锛屽氨绉垎寰楃粨鏋 绗笁棰橈細鍏堟壘鍑鸿繖涓绉垎鐨缁撴灉锛岀劧鍚庝唬鍏ƒ(x)绗洓棰橈細涓よ竟...
绛旓細鍥炵瓟锛3. 鈭玿lnxdx=x²lnx/2-鈭玿²*1/2xdx=x²lnx/2-鈭玿/2dx=x²lnx/2-x²/4+c
绛旓細瑙ｏ細鈭玠x/x(1-x)=鈭玠x/x+鈭玠x/(1-x)=鈭玠x/x-鈭玠(1-x)/(1-x)=ln|x|-ln|1-x|+C 鈭玸inx^8dx=x*sinx^8-鈭玿d(sinx^8)=x*sinx^8-8鈭玿^8*cosx^8dx=x*sinx^8-8鈭玿^8*(1-sinx^8)dx=x*sinx^8-8鈭玿^8dx+8鈭玿^8*sinx^8dx=x*sinx^8-8/9*x^9+8鈭玿^8*sinx^...
绛旓細鍥炵瓟锛氱涓棰,cosx*cos3x=cos2x+cos4x 绗簩棰,浠=2+(x-3)^1/3 绗笁棰,1-4x-x^2=(5-(x-2)^2)^1/2,鍚庨潰鐢╝rcsin鍙樻崲 绗洓棰,t=(x-1)^1/2,鍒嗛儴绉垎 绗簲棰,鍒嗘瘝涓(x+3)(x-2),鎷嗗紑
绛旓細x) + C 鎻愮ず锛氶亣鍒扳垰(1 卤 x²)绛夌瓑杩欑甯︽牴鍙鐨勯鐩鏃讹紝瑕佹崲鍏冧笁瑙掑嚱鏁版潵鍋 2銆佲埆 (x⁴ + 2x² + x + 1)/[x(1 + x²)] dx = 鈭 [x + 1/x + 1/(x² + 1)] dx = x²/2 + ln|x| + arctan(x) + C 鎻愮ず锛氱敤澶氶」寮忕患鍚堥櫎娉曘
