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设â(x+2)=tï¼åx=(t^2-2),ä»£å ¥å¾ï¼
â«xâ(x+2)dx
=â«t*(t^2-2)d(t^2-2),
=2â«t^2*(t^2-2)dt,
=2â«(t^4-2t^2)dt,
=2/5*t^5-4/3*t^3+C,
=2/5*(x+2)^(5/2)-4/3*(x+2)^(3/2)+C,
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â«xâ(x+2)dx
=â«xâ(x+2)d(x+2),
=2/3â«xd(x+2)^(3/2),
=2/3*x(x+2)^(3/2)- 2/3â«(x+2)^(3/2)dx,
=2/3*x(x+2)^(3/2)- 4/3â«(x+2)^(3/2)d(x+2),
=2/3*x(x+2)^(3/2)- 4/15*(x+2)^(5/2)+C,
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A=â«xâ(x+2)dx,
=(1/2)â«â(x+2)dx^2,
=(1/2)x^2â(x+2)-(1/2)â«x^2dâ(x+2),
=(1/2)x^2â(x+2)-(1/4)â«x^2/â(x+2)dx,
=(1/2)x^2â(x+2)-(1/4)â«[x(x+2)-2*(x+2)+4]/â(x+2)dx,
=(1/2)x^2â(x+2)-(1/4)A+1/2â«â(x+2)dx-â«dx/â(x+2),
å³ï¼(5/4)A=(1/2)x^2â(x+2)+1/2â«â(x+2)dx-2â«dx/2â(x+2),
A=(2/5)x^2â(x+2)+2/5â«â(x+2)d(x+2)-8/5â(x+2),
A=(2/5)x^2â(x+2)+4/15(x+2)^(3/2)-8/5*â(x+2)+Cã
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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
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初等函数积不出来,二重积分的方法可以得到:
[∫exp(x^2)dx]^2
=∫exp(y^2)dy∫exp(x^2)dx
=∫∫exp(x^2+y^2)dxdy
看到一个圆的表达式了,用极坐标代换
=∫∫rexp(r^2)drdθ
假设圆的半径是r
=2π[(1/2)exp(r^2)] =π[exp(a^2)-1]
因此∫exp(x^2)dx=根号下π[exp(a^2)-1])
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