搜狗做题网积分（-π/2--0）x^2 cosnx dx 怎么解
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积分（-π/2--0）x^2 cosnx dx 怎么解
当n=0时,
∫ [-π/2-->0] x²dx
=1/3x³ [-π/2-->0]
=π³/24
当n>0时,
∫ [-π/2-->0] x²cosnx dx
=(1/n)∫ [-π/2-->0] x² d(sinnx)
=(1/n)x²sinnx-(1/n)∫ [-π/2-->0] 2xsinnx dx
=(1/n)x²sinnx+(2/n²)∫ [-π/2-->0] x d(cosnx)
=(1/n)x²sinnx+(2/n²)xcosnx-(2/n²)∫ [-π/2-->0] cosnxdx
=(1/n)x²sinnx+(2/n²)xcosnx-(2/n³)sinnx [-π/2-->0]
=-(1/n)(π²/4)sin(-nπ/2)-(2/n²)(-π/2)cos(-nπ/2)-(2/n³)sin(-nπ/2)
=(1/n)(π²/4)sin(nπ/2)+(2/n²)(π/2)cos(nπ/2)+(2/n³)sin(nπ/2)
∫ [-π/2-->0] x²dx
=1/3x³ [-π/2-->0]
=π³/24
当n>0时,
∫ [-π/2-->0] x²cosnx dx
=(1/n)∫ [-π/2-->0] x² d(sinnx)
=(1/n)x²sinnx-(1/n)∫ [-π/2-->0] 2xsinnx dx
=(1/n)x²sinnx+(2/n²)∫ [-π/2-->0] x d(cosnx)
=(1/n)x²sinnx+(2/n²)xcosnx-(2/n²)∫ [-π/2-->0] cosnxdx
=(1/n)x²sinnx+(2/n²)xcosnx-(2/n³)sinnx [-π/2-->0]
=-(1/n)(π²/4)sin(-nπ/2)-(2/n²)(-π/2)cos(-nπ/2)-(2/n³)sin(-nπ/2)
=(1/n)(π²/4)sin(nπ/2)+(2/n²)(π/2)cos(nπ/2)+(2/n³)sin(nπ/2)