计算定积分∫(π/2~0) x(sinx+cosx) dx

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计算定积分∫(π/2~0) x(sinx+cosx) dx
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计算定积分∫(π/2~0) x(sinx+cosx) dx
计算定积分∫(π/2~0) x(sinx+cosx) dx

计算定积分∫(π/2~0) x(sinx+cosx) dx
∫x(sinx+cosx)dx
=∫xsinxdx+∫xcosxdx
=-∫xd(cosx)+∫xcosxdx
=-xcosx+∫cosxdx+∫xd(sinx)
=-xcosx+sinx+xsinx-∫sinxdx
=-xcosx+sinx+xsinx+cosx
∫(π/2~0) x(sinx+cosx)=(-xcosx+sinx+xsinx+cosx
)|(π/2~0)
=(1+π/2)-(1)=π/2