求定积分∫lncosx dx(o≤x≤π/4)
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∫lncosx dx(o≤x≤π/4)
=xlncosx (o≤x≤π/4)-∫xd(lncosx )(o≤x≤π/4)
=xlncosx (o≤x≤π/4)+∫xtanxdx(o≤x≤π/4)
=-πln2/8+∫xtanxdx(o≤x≤π/4)
其中右部分
∫xtanxdx(o≤x≤π/4)
==1/2积分:arctanxdx^2(o≤x≤π/4)
=x^2/2arctanx-1/2积分:x^2d(arctanx)(o≤x≤π/4)
=x^2/2arctanx-1/2积分:x^2/(1+x^2)dx(o≤x≤π/4)
=x^2/2arctanx-1/2积分:(x^2+1-1)/(1+x^2)dx(o≤x≤π/4)
=x^2/2arctanx-x/2+arctanx(o≤x≤π/4)
=π^2/(4arctan(π/4))-π/8+arctan(π/4)
=xlncosx (o≤x≤π/4)-∫xd(lncosx )(o≤x≤π/4)
=xlncosx (o≤x≤π/4)+∫xtanxdx(o≤x≤π/4)
=-πln2/8+∫xtanxdx(o≤x≤π/4)
其中右部分
∫xtanxdx(o≤x≤π/4)
==1/2积分:arctanxdx^2(o≤x≤π/4)
=x^2/2arctanx-1/2积分:x^2d(arctanx)(o≤x≤π/4)
=x^2/2arctanx-1/2积分:x^2/(1+x^2)dx(o≤x≤π/4)
=x^2/2arctanx-1/2积分:(x^2+1-1)/(1+x^2)dx(o≤x≤π/4)
=x^2/2arctanx-x/2+arctanx(o≤x≤π/4)
=π^2/(4arctan(π/4))-π/8+arctan(π/4)
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