求解道定积分题,要详解,
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求解道定积分题,要详解,求解道定积分题,要详解,求解道定积分题,要详解,∫(0,1)(√(1-(1-x)^2)-x)dx令1-x=cost,则√(1-(1-x)^2)=sint,x=1-cost,dx
求解道定积分题,要详解,
求解道定积分题,要详解,
求解道定积分题,要详解,
∫(0,1)(√(1-(1-x)^2)-x)dx
令1-x=cost,则√(1-(1-x)^2)=sint,x=1-cost,dx=sintdt,
原式=∫(0, π/2)（sint-1+cost）sintdt=-π/2+∫(0, π/2)sint^2dt+∫(0, π/2)sintcostdt
=-π/2+1/2*∫(0, π/2)(1-cos2t)dt+1/2*∫(0, π/2)sin2tdt
=-π/2+1/2*(t-1/2*sin2t)+1/2*(-1/2*cos2t) (0, π/2)
= -π/2+1/2*(π/2-1/2*(-1+1))
= -π/2+π/4
=-π/4