integral from sqrt(y) to 1 of e^{x^3}

解答

\int_{y}^{1} e^{x^{3}} d x

- \frac{1 \cdot Γ ( \frac{1}{3} , - 1 ^{3} )}{3 3 - 1 ^{3}} - - \frac{y Γ ( \frac{1}{3} , - ( y ) ^{3} )}{3 3 - ( y ) ^{3}}

求解步骤

\int_{y}^{1} 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}}]_{y}^{1}

带入上下限计算后得:

- \frac{1 \cdot Γ ( \frac{1}{3} , - 1 ^{3} )}{3 3 - 1 ^{3}} - - \frac{y Γ ( \frac{1}{3} , - ( y ) ^{3} )}{3 3 - ( y ) ^{3}}

= - \frac{1 \cdot Γ ( \frac{1}{3} , - 1 ^{3} )}{3 3 - 1 ^{3}} - - \frac{y Γ ( \frac{1}{3} , - ( y ) ^{3} )}{3 3 - ( y ) ^{3}}

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