integral from 0 to pi/4 of cos(x^3)

解

\int_{0}^{\frac{π}{4}} cos (x^{3}) d x

- \frac{\frac{π}{4} ( 3 i ( \frac{π}{4} ) ^{3} Γ ( \frac{1}{3} , - i ( \frac{π}{4} ) ^{3} ) + 3 - i ( \frac{π}{4} ) ^{3} Γ ( \frac{1}{3} , i ( \frac{π}{4} ) ^{3} ) )}{6 3 ( \frac{π}{4} ) ^{6}} - - \frac{0 \cdot ( 3 i 0 ^{3} Γ ( \frac{1}{3} , - i 0 ^{3} ) + 3 - i 0 ^{3} Γ ( \frac{1}{3} , i 0 ^{3} ) )}{6 3 0 ^{6}}

解答ステップ

\int_{0}^{\frac{π}{4}} cos (x^{3}) d x

非初等積分を使用する:

\int cos (x^{3}) d x = - \frac{x ( 3 i x ^{3} Γ ( \frac{1}{3} , - i x ^{3} ) + 3 - i x ^{3} Γ ( \frac{1}{3} , i x ^{3} ) )}{6 3 x ^{6}}

= - \frac{x ( 3 i x ^{3} Γ ( \frac{1}{3} , - i x ^{3} ) + 3 - i x ^{3} Γ ( \frac{1}{3} , i x ^{3} ) )}{6 3 x ^{6}}_{0}^{\frac{π}{4}}

限度を計算する:

- \frac{\frac{π}{4} ( 3 i ( \frac{π}{4} ) ^{3} Γ ( \frac{1}{3} , - i ( \frac{π}{4} ) ^{3} ) + 3 - i ( \frac{π}{4} ) ^{3} Γ ( \frac{1}{3} , i ( \frac{π}{4} ) ^{3} ) )}{6 3 ( \frac{π}{4} ) ^{6}} - - \frac{0 \cdot ( 3 i 0 ^{3} Γ ( \frac{1}{3} , - i 0 ^{3} ) + 3 - i 0 ^{3} Γ ( \frac{1}{3} , i 0 ^{3} ) )}{6 3 0 ^{6}}

= - \frac{\frac{π}{4} ( 3 i ( \frac{π}{4} ) ^{3} Γ ( \frac{1}{3} , - i ( \frac{π}{4} ) ^{3} ) + 3 - i ( \frac{π}{4} ) ^{3} Γ ( \frac{1}{3} , i ( \frac{π}{4} ) ^{3} ) )}{6 3 ( \frac{π}{4} ) ^{6}} - - \frac{0 \cdot ( 3 i 0 ^{3} Γ ( \frac{1}{3} , - i 0 ^{3} ) + 3 - i 0 ^{3} Γ ( \frac{1}{3} , i 0 ^{3} ) )}{6 3 0 ^{6}}