sinx和cosx多次方乘积的积分.docx

sinx和cosx多次方乘积的积分我们要求的是在不同区间内求$SinX$和$COSX$的多次方乘积的积分。首先，我们需要确定这些积分所在的区间。1 .当$*111-fracpi2,fracpi4$时,$sinxge0$,$cosxge0$,所以$sinxcdotcosX$在$-fracpi2,fracpi4$上连续。2 .当$*1.11红2(514,frac3pi4$时，$sinxge0$,$cosxle0$,所以$sinxcdotcosX$在$fracpi4,frac3pi4$上连续。3 .Sxin-frac3pi4,-fracpi2$时,$sinxle0$,$cosxle0$,所以$sinxcdotcosX$在$-力?(3pi4,-fracpi2$上连续。接下来，我们分别计算这三个区间内的积分：1 .在$-fracpi2,fracpi4$上的积分：$int_-fracpi2fracpi4sinxdx=(sin-cosx)|_fracpi2Afracpi4二(sinfracpi4-cosfracpi4)一(sin(-fracpi2)-cos(-fracpi2)=(sqrt2)2+1$.2 .在$&°力14,frac3pi4$上的积分：$int_fracpi4-frac3pi4sinxdx=(sin-cosx)|_fracpi4frac3pi4二(sinfrac3pi4-cosfrac3pi4)-(sinfracpi4-cosfracpi4)=(sqrt2)2+1$.3 .在$-frac3pi4,-fracpi2$上的积分:$int_-dfrac3pi4-fracpi2sinxdx=(sin-cosx)|_-frac3pi4-fracpi2二(sin(-dfracpi2)-cos(-dfracpi2)-(sin(-dfrac3pi4)-cos(-dfrac3pi4)=-1+dfracsqrt22$.最后，将这三个积分相加，得到$后1-这三y+infty)Sinxcosxdx=(sqrt2)/2+1+(sqrt2)2+1-1+dfracsqrt22=dfrac3(sqrt2)2+1$.