求不定积分1
\u6c42\u4e0d\u5b9a\u79ef\u5206\u222b1/xdx\u7b54\uff1a
\u56e0\u4e3a\u79ef\u5206\u51fd\u6570y=f(x)=1/x\u662f\u53cd\u6bd4\u4f8b\u51fd\u6570\uff0c\u5b58\u5728\u4e24\u652f
\u6240\u4ee5\uff1ax0\u90fd\u8981\u8003\u8651
x>0\u65f6\u79ef\u5206\u5f97\uff1alnx+C
x<0\u65f6\uff1a
\u222b 1/x dx=\u222b 1/(-x) d(-x)=ln(-x)+C
\u7efc\u4e0a\u6240\u8ff0\uff0c\u222b1/x dx=ln|x|+C
x<0\u65f6\uff0cln(-x)\u7684\u5bfc\u6570\u4e5f\u662f1/x
\u7b54\uff1a
\u222b 1/[x(1+x^2)] dx
=\u222b 1/x - x/(1+x^2) dx
=lnx-(1/2)ln(1+x^2)+C
绛旓細鈭 1/[x(x^10+1)] dx = 鈭 [(x^10+1)-x^10]/[x(x^10+1)] dx = 鈭 1/x dx - 鈭 x^9/(x^10+1) dx = 鈭 1/x dx - (1/10)鈭 d(x^10+1)/(x^10+1)= ln|x| - (1/10)ln|x^10 + 1| + C 杩炵画鍑芥暟锛屼竴瀹氬瓨鍦ㄥ畾绉垎鍜涓嶅畾绉垎锛涜嫢鍦ㄦ湁闄愬尯闂碵a,b]涓...
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绛旓細绛旓細鍥犱负绉垎鍑芥暟y=f(x)=1/x鏄弽姣斾緥鍑芥暟锛屽瓨鍦ㄤ袱鏀 鎵浠ワ細x<0鍜寈>0閮借鑰冭檻 x>0鏃剁Н鍒嗗緱锛歭nx+C x<0鏃讹細鈭 1/x dx=鈭 1/(-x) d(-x)=ln(-x)+C 缁间笂鎵杩帮紝鈭1/x dx=ln|x|+C x<0鏃讹紝ln(-x)鐨勫鏁颁篃鏄1/x
绛旓細(1)鈭玿²e^(-2x)dx =(-½)鈭玿²d[e^(-2x)]=(-½)x²路e^(-2x) +½鈭玔e^(-2x)]d(x²)=(-½)x²路e^(-2x) +½鈭玔2x路e^(-2x)]dx =(-½)x²路e^(-2x) +鈭玔x路e^(-2x)]dx =(-½)x...
绛旓細1/(sinx+cosx)鐨涓嶅畾绉垎鍏蜂綋鍥炵瓟濡備笅锛氣埆1/(sinx+cosx) dx =鈭1/[鈭2(sinxcos蟺/4+sin蟺/4路cosx)]dx =鈭1/[鈭2sin(x+蟺/4)] dx =鈭2/2 鈭玞sc(x+蟺/4) d(x+蟺/4)=鈭2/2 ln|csc(x+蟺/4)-cot(x+蟺/4)|+C 涓嶅畾绉垎鐨勫叕寮忥細1銆佲埆adx=ax+C锛宎鍜孋閮芥槸甯告暟 2銆...
绛旓細鍘熷紡=鈭玜secz*asec²z dz =鈭玸ecz dtanz,a²鍏堢渷鐣 =secztanz - 鈭玹anz dsecz =secztanz - 鈭玹anz(secztanz) dz =secztanz - 鈭玸ec³z dz + 鈭玸ecz dz 鈭2鈭玸ec³z dz = secztanz + ln|secz + tanz| 鈭粹埆sec³z dz = (1/2)secztanz + (...
绛旓細1.=鈭紙1锛2x锛1/8dx=1/2鈭紙1锛2x锛1/8d锛1锛2x锛=1/2脳锛1锛2x锛1/8锛1/锛1/8锛1锛=4/9锛1锛2x锛9/8
绛旓細璁3x+4=t x=锛坱-4锛/3 鍘熷紡=鈭紙t-4锛/9t^2dt =1/9鈭1/t-4/t^2dt =1/9ln|t|+4/(9t)+C 缁撴灉涓1/9ln|3x+4|+4/9锛3x+4锛+C
绛旓細鍋囪鏄1/sinx鐨绉垎 鈭1/sinx dx =鈭1/[2sin(x/2)cos(x/2)] dx,涓ゅ嶈鍏紡 =鈭1/[sin(x/2)cos(x/2)] d(x/2)=鈭1/tan(x/2)*sec²(x/2) d(x/2)=鈭1/tan(x/2) d[tan(x/2)], [娉ㄢ埆sec²(x/2)d(x/2)=tan(x/2)+C]=ln|tan(x/2)|+C, ...