â« (sinx)^4dx=(sin4x)/32 - (sin2x)/4 + (3x/8) + CãC为积å常æ°ã
解çè¿ç¨å¦ä¸ï¼
(sinx)^4
= (sinx^2)^2
= ((1 - cos2x)/2)^2
= (1 - 2cos2x + (cos2x)^2)/4
= 0.25 - 0.5cos2x + 0.125(1 + cos4x)
= (cos4x)/8 - (cos2x)/2 + 3/8
â« (sinx)^4dx
= â« ((cos4x)/8 - (cos2x)/2 + 3/8)dx
= â« ((cos4x)/8)dx - â« ((cos2x)/2)dx + â« (3/8)dx
= (1/32)â« cos4xd4x - (1/4)â« cos2xd2x + (3x/8)
= (sin4x)/32 - (sin2x)/4 + (3x/8) + C
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(uv)'=u'v+uv'
å¾ï¼u'v=(uv)'-uv'
两边积åå¾ï¼â« u'v dx=â« (uv)' dx - â« uv' dx
å³ï¼â« u'v dx = uv - â« uv' d,è¿å°±æ¯åé¨ç§¯åå ¬å¼
ä¹å¯ç®å为ï¼â« v du = uv - â« u dv
ä¸å®ç§¯åçå ¬å¼
1ãâ« a dx = ax + Cï¼aåCé½æ¯å¸¸æ°
2ãâ« x^a dx = [x^(a + 1)]/(a + 1) + Cï¼å ¶ä¸a为常æ°ä¸ a â -1
3ãâ« 1/x dx = ln|x| + C
4ãâ« a^x dx = (1/lna)a^x + Cï¼å ¶ä¸a > 0 ä¸ a â 1
5ãâ« e^x dx = e^x + C
6ãâ« cosx dx = sinx + C
7ãâ« sinx dx = - cosx + C
8ãâ« cotx dx = ln|sinx| + C = - ln|cscx| + C