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)