求不定积分(最好有过程):1.∫e^x/x dx 2.∫(sinx)^(-1/2) dx

2个回答

  • 1、∫ (e^x)/x dx = Ei(x) + C,指数积分的定义

    或用级数

    ∫ (e^x)/x dx

    = ∫ 1/x · Σ(k=0→∞) (x^k)/k!dx

    = ∑(k=0→∞) 1/k!· ∫ x^(k - 1) dx

    = ∑(k=0→∞) 1/k!· x^(k - 1 + 1)/(k - 1 + 1) + C

    = ∑(k=0→∞) (x^k)/(k!· k) + C

    2、这个是椭圆积分,给定上下限π/2和0

    ∫(0→π/2) (sinx)^(- 1/2) dx

    = ∫(0→π/2) 1/√(sinx) dx

    -->令x = π/2 - u-->

    = ∫(0→π/2) 1/√(cosu) du

    -->令cosu = cos²z,- sinu du = - 2coszsinz dz,du = 2cosz/√(1 + cos²z) dz-->

    = 2∫(0→π/2) 1/√(1 + cos²z) dz

    = 2∫(0→π/2) 1/√(2 - sin²z) dz

    = √2∫(0→π/2) 1/√[1 - (1/2)sin²z] dz

    = √2F(1/√2,π/2) 或 √2K(1/√2)