integral from 0 to 0.5 of e^{-x^3}

解答

\int_{0}^{0.5} e^{- x^{3}} d x

- \frac{0.5Γ ( \frac{1}{3} , 0. 5 ^{3} )}{3 3 0. 5 ^{3}} - (- \frac{0 \cdot Γ ( \frac{1}{3} , 0 ^{3} )}{3 3 0 ^{3}})

求解步骤

\int_{0}^{0.5} e^{- x^{3}} d x

这是非初等函数积分:

\int e^{- x^{a}} d x = - \frac{x Γ ( \frac{1}{a} , x ^{a} )}{a a x ^{a}}

= [- \frac{x Γ ( \frac{1}{3} , x ^{3} )}{3 3 x ^{3}}]_{0}^{0.5}

带入上下限计算后得:

- \frac{0.5Γ ( \frac{1}{3} , 0. 5 ^{3} )}{3 3 0. 5 ^{3}} - (- \frac{0 \cdot Γ ( \frac{1}{3} , 0 ^{3} )}{3 3 0 ^{3}})

= - \frac{0.5Γ ( \frac{1}{3} , 0. 5 ^{3} )}{3 3 0. 5 ^{3}} - (- \frac{0 \cdot Γ ( \frac{1}{3} , 0 ^{3} )}{3 3 0 ^{3}})

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