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∫sinxsin2xsin3xDx

这题利用公式求

积化和差∫sinxsin2xsin3xdx=1/2∫(cosx-cos3x)sin3xdx=1/2∫cosxsin3xdx-1/2∫cos3xsin3xdx=1/4∫(sin2x+sin4x)dx-1/4∫sin6xdx=-1/8cos2x-1/16cos4x+1/24cos6x+C数学软件验算:

被积函数是奇函数,而积分区间关于原点对称,根据”奇函数在对称区间的积分为0“可知,原式=0.

因为2sin5xcos3x=(sin5xcos3x+cos5xsin3x)+(sin5xcos3x-cos5xsin3x)=sin8x+sin2x 也可以用积化和差公式 sinxcosy=(1/2)*(sin(x+y)+sin(x-y))

∫ xcos2x.sinx dx =-∫ xcos2x.dcosx =-xcos2x.cosx +∫ cosx.( cos2x - 2xsin2x ) dx =-xcos2x.cosx +∫ cosx.cos2x dx - 2∫xsin2x.cosx dx =-xcos2x.cosx +(1/2)∫ (cosx+ cos3x) dx - 2∫xsin2x dsinx =-xcos2x.cosx +(1/2)sinx+ (1/6)sin3x - 2[x...

证明:由题意,P=2xcosy-y2sinx,Q=2ycosx-x2siny,在整个平面上具有一阶连续偏导数,且?P?y=?2xsiny?2ysinx=?Q?x∴曲线积分I与积分路径无关.取路径从(0,0)到(2,0)再到(0,3),则I=∫202xdx+∫30(2ycos2?4siny)dy=4+9cos2+4cos3-4=9cos...

用分步积分 ∫(xcosx)/(sinx)^3 dx =∫(x)/(sinx)^3 dsinx =-1/2∫(x) d(1/sin^2x) =-1/2s/sin^2x+1/2∫1/sin^2xdx =-1/2s/sin^2x+1/2∫csc^2xdx =-1/2s/sin^2x-1/2cotx+C

∫[0,π]x^2(sinx^2)dx =∫[0,π]x^2(1/2)(1-cos2x)dx =∫[0,π](1/2)x^2dx-∫[0,π](1/2)x^2cos2xdx =(1/6)π^3 -(1/4)∫[0,π]x^2dsin2x =(1/6)π^3+(1/2)∫[0,π]xsin2xdx =(1/6)π^3+(-1/4)∫[0,π]xdcos2x =(1/6)π^3+(-1/4)π+(1/4)∫[0,π]cos2xdx =(1/6)π^3+(...

两种方法 (1)积化和差公式 ∫sinxsin2xsin3xdx =1/2·∫(cosx-cos3x)sin3xdx =1/2·∫(cosx·sin3x-cos3x·sin3x)dx =1/4·∫(sin4x-...

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