ä¾å¦è®¡ç®ä¸å®ç§¯åâ«x²3â1-xdx
解ï¼åå¼=3â«x²â1-x
令â1-x=t
x=1-t²
dx=-2tdt
åå¼=3â«ï¼1-t²ï¼²t(-2t)dt
=3â«ï¼-2t²+4t^4-2t^6ï¼dt
=-6â«t²dt+12â«t^4dt-6â«t^6dt
=-2t^3+12/5t^5-6/7t^7+c
=-2â(1-x)^3+12/5â(1-x)^5-6/7â(1-x)^7+cã
ä¾å¦æ¬é¢ä¸å®ç§¯å计ç®è¿ç¨å¦ä¸ï¼
â«ï¼1-3xï¼^6dx
=(-1/3)â«(1-3x)^6d(1-3x)
=-1/3*(1-3x)^7*(1/7)+C
=-1/21*ï¼1-3xï¼^7+Cã
ä¾å¦ä¸å®ç§¯åâ«1/(2+ cosx)计ç®
设t=tan(x/2)
åcosx=[cos²(x/2)-sin²(x/2)]/[cos²(x/2)+sin²(x/2)]
=[1-tan²(x/2)]/[1+tan²(x/2)]
=(1-t²)/(1+t²)
dx=d(2arctant)=2dt/(1+t²)
æ ï¼â«1/(2+cosx)dx=â«1/[2+(1-t²)/(1+t²)]*[2dt/(1+t²)]
=â«2dt/(3+t²)
=2/â3â«d(t/â3)/[1+(t/â3)²]
=2/â3arctan(t/â3)+C
åä¾å¦â«lntanx/(sinxcosx)dx
åååæ¯åé¤ä»¥cos²x
=â«sec²x*lntanx/tanxdx
=â«lntanx/tanx d(tanx)
=â«lntanxd(lntanx)
=(1/2)ln²(tanx)+Cã
æ¢å æ³è®¡ç®ä¸å®ç§¯å
ä¾å¦â« â(x²+1) dx
令x=tanuï¼åâ(x²+1)=secuï¼dx=sec²uduã
â«sec³udu
=â« secudtanu
=secutanu - â« tan²usecudu
=secutanu - â« (sec²u-1)secudu
=secutanu - â« sec³udu + â« secudu
=secutanu - â« sec³udu + ln|secu+tanu|
å°- â« sec³udu移æ¯çå¼å·¦è¾¹ä¸å·¦è¾¹å并åé¤ä»¥ç³»æ°å¾ï¼
â«sec³udu=(1/2)secutanu + (1/2)ln|secu+tanu| + Cã
æ以ï¼
â« â(x²+1) dx=(1/2)â(x²+1)*x+ (1/2)ln|â(x²+1)+x| + Cã
ä¸å®ç§¯åæ¦å¿µ
设F(x)æ¯å½æ°f(x)çä¸ä¸ªåå½æ°ï¼æ们æå½æ°f(x)çææåå½æ°F(x)+ C(å ¶ä¸ï¼C为任æ常æ°ï¼å«åå½æ°f(x)çä¸å®ç§¯åï¼åå«åå½æ°f(x)çå导æ°ï¼è®°ä½â«f(x)dxæè â«fï¼é«ç微积åä¸å¸¸çå»dxï¼ï¼å³â«f(x)dx=F(x)+Cã
å ¶ä¸â«å«å积åå·ï¼f(x)å«å被积å½æ°ï¼xå«å积ååéï¼f(x)dxå«å被积å¼ï¼Cå«å积å常æ°æ积å常éï¼æ±å·²ç¥å½æ°çä¸å®ç§¯åçè¿ç¨å«å对è¿ä¸ªå½æ°è¿è¡ä¸å®ç§¯åã
可以用反函数来做
y=arccosx,
∫arccosxdx=∫ydcosy=ycosy-∫cosydy
=ycosy-siny+C
=xarccosx-√(1-x^2)+C
不定积分的公式
1、∫ a dx = ax + C,a和C都是常数
2、∫ x^a dx = [x^(a + 1)]/(a + 1) + C,其中a为常数且 a ≠ -1
3、∫ 1/x dx = ln|x| + C
4、∫ a^x dx = (1/lna)a^x + C,其中a > 0 且 a ≠ 1
5、∫ e^x dx = e^x + C
6、∫ cosx dx = sinx + C
7、∫ sinx dx = - cosx + C
8、∫ cotx dx = ln|sinx| + C = - ln|cscx| + C