x的四次方乘sinx的三次方的积分
来源:学生作业帮 编辑:作业帮 分类:数学作业 时间:2024/05/09 08:14:22
x的四次方乘sinx的三次方的积分
∫x^4(sinx)^3dx
=(1/4)∫x^4(3sinx-sin3x)dx
=(3/4)∫x^4sinxdx-(1/12)∫x^4sin3xd(3x)
=-(3/4)∫x^4d(cosx)+(1/12)∫x^4d(cos3x)
=-(3/4)x^4cosx+(3/4)∫cosxd(x^4)+(1/12)x^4cos3x-(1/12)∫cos3xd(x^4)
=-(3/4)x^4cosx+(1/12)x^4cos3x+3∫x^3cosxdx-(1/3)∫x^3cos3xdx
=(1/12)x^4cos3x-(3/4)x^4cosx+3∫x^3d(sinx)-(1/9)∫x^3d(sin3x)
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-3∫sinxd(x^3)-(1/9)x^3sin3x
+(1/9)∫sin3xd(x^3)
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x-9∫x^2sinxdx
+(1/3)∫x^2sin3xdx
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9∫x^2d(cosx)
-(1/9)∫x^2d(cos3x)
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9x^2cosx-9∫cosxd(x^2)
-(1/9)x^2cos3x+(1/9)∫cos3xd(x^2)
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9x^2cosx
-(1/9)x^2cos3x-18∫xcosxdx+(2/9)∫xcos3xdx
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9x^2cosx
-(1/9)x^2cos3x-18∫xd(sinx)+(2/27)∫xd(sin3x)
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9x^2cosx
-(1/9)x^2cos3x-18xsinx+18∫sinxdx+(2/27)xsin3x-(2/27)∫sin3xdx
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9x^2cosx
-(1/9)x^2cos3x-18xsinx-18cosx+(2/27)xcos3x+(2/81)cos3x+C.
=(1/4)∫x^4(3sinx-sin3x)dx
=(3/4)∫x^4sinxdx-(1/12)∫x^4sin3xd(3x)
=-(3/4)∫x^4d(cosx)+(1/12)∫x^4d(cos3x)
=-(3/4)x^4cosx+(3/4)∫cosxd(x^4)+(1/12)x^4cos3x-(1/12)∫cos3xd(x^4)
=-(3/4)x^4cosx+(1/12)x^4cos3x+3∫x^3cosxdx-(1/3)∫x^3cos3xdx
=(1/12)x^4cos3x-(3/4)x^4cosx+3∫x^3d(sinx)-(1/9)∫x^3d(sin3x)
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-3∫sinxd(x^3)-(1/9)x^3sin3x
+(1/9)∫sin3xd(x^3)
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x-9∫x^2sinxdx
+(1/3)∫x^2sin3xdx
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9∫x^2d(cosx)
-(1/9)∫x^2d(cos3x)
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9x^2cosx-9∫cosxd(x^2)
-(1/9)x^2cos3x+(1/9)∫cos3xd(x^2)
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9x^2cosx
-(1/9)x^2cos3x-18∫xcosxdx+(2/9)∫xcos3xdx
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9x^2cosx
-(1/9)x^2cos3x-18∫xd(sinx)+(2/27)∫xd(sin3x)
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9x^2cosx
-(1/9)x^2cos3x-18xsinx+18∫sinxdx+(2/27)xsin3x-(2/27)∫sin3xdx
=(1/12)x^4cos3x-(3/4)x^4cosx+3x^3sinx-(1/9)x^3sin3x+9x^2cosx
-(1/9)x^2cos3x-18xsinx-18cosx+(2/27)xcos3x+(2/81)cos3x+C.
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