求解不定积分题目 求解一道不定积分的题目
\u6c42\u89e3\u4e00\u4e2a\u4e0d\u5b9a\u79ef\u5206\u9898\u76ee\uff08\u8981\u4e2d\u95f4\u6b65\u9aa4\uff09\u222b\u221a(2x-x^2)dx
=\u222b\u221a(1-(x-1)^2)d(x-1)
\u4ee4t=x-1 \u66ff\u6362
=\u222b\u221a(1-t^2)dt
\u5c06t=sinu \u66ff\u6362
=\u222b\u221a(1-sinu^2)*cosudu
=\u222b(cosu)^2du
=1/2\u222b(1+cos2u)du
=1/2\u222bdu +1/4\u222bcos2ud2u
=1/2u +1/4sin2u +C
u=arcsint=arcsin(x-1)
\u4ee3\u5165\u5f97\u5230
=1/2arcsin(x-1)+1/2(x-1)+C
\u4ee4x\uff1d2tanu\uff0c\u5219\uff1a
tanu\uff1dx/2\u3001u\uff1darctan\uff08x/2\uff09\uff0c
sinu\uff1dtanu/\u221a\uff3b1\uff0b\uff08tanu\uff09^2\uff3d\uff1d\uff08x/2\uff09/\u221a\uff3b1\uff0b\uff08x/2\uff09^2\uff3d\uff1dx/\u221a\uff084\uff0bx^2\uff09\uff0c
dx\uff1d\uff3b2/\uff08cosu\uff09^2\uff3ddu\u3002
\u2234\u222b\uff3b\u221a\uff08x^2\uff0b4\uff09/x^2\uff3ddx
\uff1d\u222b\uff5b\u221a\uff3b\uff082tanu\uff09^2\uff0b4\uff3d/\uff082tanu\uff09^2\uff5d\uff3b2/\uff08cosu\uff09^2\uff3ddu
\uff1d\u222b\uff3b\uff081/cosu\uff09/\uff08tanu\uff09^2\uff3d\uff3b1/\uff08cosu\uff09^2\uff3ddu
\uff1d\u222b\uff5b1/\uff3bcosu\uff08sinu\uff09^2\uff3d\uff5ddu
\uff1d\u222b\uff5bcosu/\uff3b\uff08cosu\uff09^2\uff08sinu\uff09^2\uff3d\uff5ddu
\uff1d\u222b\uff5b1/\uff3b\uff08sinu\uff09^2\uff0d\uff08sinu\uff09^4\uff3d\uff5dd\uff08sinu\uff09
\uff1d\uff081/2\uff09\u222b\uff5b1/\uff3b1\uff0d\uff08sinu\uff09^2\uff3d\uff0b1/\uff3b1\uff0b\uff08sinu\uff09^2\uff3d\uff5dd\uff08sinu\uff09
\uff1d\uff081/2\uff09\u222b\uff5b1/\uff3b1\uff0d\uff08sinu\uff09^2\uff3d\uff5dd\uff08sinu\uff09\uff0b\uff081/2\uff09\u222b\uff5b1/\uff3b1\uff0b\uff08sinu\uff09^2\uff3d\uff5dd\uff08sinu\uff09
\uff1d\uff081/4\uff09\u222b\uff3b1/\uff081\uff0dsinu\uff09\uff0d1/\uff081\uff0bsinu\uff09\uff3dd\uff08sinu\uff09\uff0b\uff081/2\uff09arctan\uff08sinu\uff09
\uff1d\uff0d\uff081/4\uff09\uff3bln\uff081\uff0dsinu\uff09\uff0bln\uff081\uff0bsinu\uff09\uff3d\uff0b\uff081/2\uff09arctan\uff08sinu\uff09\uff0bC\u3002
\uff1d\uff081/2\uff09arctan\uff08sinu\uff09\uff0d\uff081/4\uff09ln\uff3b1\uff0d\uff08sinu\uff09^2\uff3d\uff0bC
\uff1d\uff081/2\uff09arctan\uff3bx/\u221a\uff084\uff0bx^2\uff09\uff3d\uff0d\uff081/4\uff09ln\uff5b1\uff0d\uff3bx/\u221a\uff084\uff0bx^2\uff09\uff3d^2\uff5d\uff0bC
\uff1d\uff081/2\uff09arctan\uff3bx/\u221a\uff084\uff0bx^2\uff09\uff3d\uff0d\uff081/4\uff09ln\uff3b4/\uff084\uff0bx^2\uff09\uff3d\uff0bC
\uff1d\uff081/2\uff09arctan\uff3bx/\u221a\uff084\uff0bx^2\uff09\uff3d\uff0d\uff081/4\uff09ln4\uff0b\uff081/4\uff09ln\uff084\uff0bx^2\uff09\uff0bC
\uff1d\uff081/2\uff09arctan\uff3bx/\u221a\uff084\uff0bx^2\uff09\uff3d\uff0b\uff081/4\uff09ln\uff084\uff0bx^2\uff09\uff0bC\u3002
简单计算一下，答案如图所示
∫arctane^xdx/e^(2x) = ∫e^x arctane^xdx/e^(3x)
= ∫arctane^xde^x/e^(3x) u = e^x
= ∫arctanudu/u^3 = (-1/2)∫arctanudu^(-2)
= (-1/2){u^(-2)arctanu -∫du/[u^2(1+u^2)]}
= (-1/2){u^(-2)arctanu -∫[1/u^2-1/(1+u^2)]du}
= (-1/2){arctanu/u^2 +1/u + arctanu} + C
= (-1/2)[arctan(e^u)/e^(2x) +1/e^x + arctan(e^x)] + C
绛旓細(1)鈭2x.e^xdx =2鈭玿.de^x =2x.e^x -2鈭玡^x dx =2x.e^x -2e^x + C (2)鈭2x.sinxdx =-2鈭玿dcosx =-2xcosx +2鈭玞osxdx =-2xcosx +2sinx + C (3)鈭(x+1)^3 dx =鈭(x+1)^3 d(x+1)=(1/4)(x+1)^4 + C (4)鈭 4dx/[1+鈭(2x) ]let 鈭(2x) = (...
绛旓細绗竴涓涓嶅畾绉垎鐨勮绠楋細\int\frac{\arctan x}{x^2}dx=\int\arctan xd(-\frac{1}{x})=-\frac{\arctan x}{x}+\int\frac{1}{x(1+x^2)}dx =-\frac{\arctan x}{x}+\int\left(\frac{1}{x}-\frac{x}{1+x^2}\right)dt =-\frac{\arctan x}{x}+\ln|x|-\frac{1...
绛旓細锛濃垰2/2ln|(tanx/2锛1锛嬧垰2)/(tanx/2锛1锛嶁垰2)锛婥 涓嶅畾绉垎鐨勬剰涔夛細濡傛灉f(x)鍦ㄥ尯闂碔涓婃湁鍘熷嚱鏁锛屽嵆鏈変竴涓嚱鏁癋(x)浣垮浠绘剰x鈭圛锛岄兘鏈塅'(x)=f(x)锛岄偅涔堝浠讳綍甯告暟鏄剧劧涔熸湁[F(x)+C]'=f(x)锛屽嵆瀵逛换浣曞父鏁癈锛屽嚱鏁癋(x)+C涔熸槸f(x)鐨勫師鍑芥暟銆傝繖璇存槑濡傛灉f(x)鏈変竴涓師鍑芥暟,...
绛旓細瑙ｇ瓟杩囩▼濡備笅锛氳繖閬撻鐢ㄤ笁瑙掍唬鎹㈡妸x鎹负3sint锛屼粠鑰宒x=d锛3sint锛=3cost锛屾墍浠ユ牴鍙9-x骞虫柟涓嶅畾绉垎灏卞彲浠ュ寲涓9cos^2t姹備笉瀹氱Н鍒嗐傝屾牴鎹2cos^2-1绛変簬cos2t锛屽彲浠ュ皢cos^2t绛変簬1/2锛坈os2t+1锛夛紝浠庤屽師寮忓氨鍙樻垚瀵9/2锛坈os2t+1锛夋眰涓嶅畾绉垎銆傝繖灏卞彲浠ュ垎鍒9/2cos2t鍜9/2姹備笉瀹氱Н鍒嗐9/...
绛旓細鈭2x/x^2-2x+5 =鈭玔(2x-2)/(x^2-2x+5)+2/(x^2-2x+5)]dx =鈭玠(x^2-2x+5)/x^2-2x+5)+2鈭玠[(x-1)/2]/{[(x-1)/2]^2+1} =ln(x^2-2x+5)+arctan[(x-1)/2]+c
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涓嶅畾绉垎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
绛旓細姹傝В涓嶅畾绉垎棰樼洰銆傝繖涓棶棰樻槸涓ソ闂锛
绛旓細=xln(x^2+1) - 2x +2arctanx +C (2)鈭玪n(lnx)/x dx =鈭玪n(lnx) dlnx =lnx .ln(lnx) - 鈭 dx/x =lnx .ln(lnx) - ln|x| +C (3)鈭玿/(cosx)^2 dx =鈭玿(secx)^2 dx =鈭玿 dtanx =xtanx - 鈭玹anx dx =xtanx + ln|cosx| +C (4)鈭(1/x^3) e^(1/x) ...
绛旓細鈭玿f'(x) dx = ln(1+x^2) +c 涓よ竟姹傚 xf'(x) = 2x/(1+x^2)f'(x) = 2/(1+x^2)f(x) =鈭 2/(1+x^2) dx = 2arctanx + C'f(1)=2 2=2(蟺/4) +C'C'= 2-蟺/2 f(x) = 2arctanx +2 -蟺/2 ...
绛旓細/(1-t³)鈭村師寮=鈭玔6t²dt/(1-t³)²]/{[2/(1-t³)]^(2/3)*[2t³/(1-t³)]^(4/3)} =鈭(6t²dt)/[2²t^4]=3/2鈭玠t/t²=(-3/2)/t+C =(-3/2)[(x+1)/(x-1)]^(1/3)+C (C鏄绉垎甯告暟)鎴栬 ...
