计算定积分
1 ∫（0→1）[(e^x)1/2]/[(e^x+e^-x)1/2]dx 2 ∫(0→1)te^(-t^2/2)dt 3 ∫(1→3^1/2) 1/[x(1+x^2)^1/2] 4 ∫(2^(1/2)/2→1) [(1-x^2)^1/2]/x^2dx
1 ∫（0→1）[(e^x)1/2]/[(e^x+e^-x)1/2]dx 实在是没有看懂这两个1/2是什么意思！ 2 ∫(0→1)te^(-t^2/2)dt =-∫(0→1)e^(-t^2/2)d(-t^2/2) =-[e^(-t^2/2)]|(0,1) =-[e^(-1/2)-1] =1-e^(-1/2) 3 ∫(1→3^1/2) 1/[x(1+x^2)^1/2] 用换元法，令x=tanθ，θ∈[π/4,π/3] 且，dx=d(tanθ)=sec^2 θdθ 原式=∫(π/4→π/3) sec^2 θdθ/[tanθ*secθ] =∫(π/4→π/3) secθdθ/tanθ =∫(π/4→π/3) dθ/sinθ =ln|cscθ-cotθ||(π/4→π/3) =ln[(√6+√3)/3] 4 ∫(2^(1/2)/2→1) [(1-x^2)^1/2]/x^2dx 用换元法，令x=sinθ，θ∈[π/4,π/2] 且，dx=d(sinθ)=cosθdθ 原式=∫(π/4→π/2) cos^2 θdθ/sin^2 θ =∫(π/4→π/2) dθ/tan^2 θ =∫(π/4→π/2) csc^2 θdθ =[-cotθ](π/4→π/2) =1。
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