integral from cos(x) to 5x of cos(u^3)

解答

\int_{c o s (x)}^{5 x} cos (u^{3}) d u

- \frac{5 x ( 3 i ( 5 x ) ^{3} Γ ( \frac{1}{3} , - i ( 5 x ) ^{3} ) + 3 - i ( 5 x ) ^{3} Γ ( \frac{1}{3} , i ( 5 x ) ^{3} ) )}{6 3 ( 5 x ) ^{6}} - - \frac{cos ( x ) ( 3 i cos ^{3} ( x ) Γ ( \frac{1}{3} , - i cos ^{3} ( x ) ) + 3 - i cos ^{3} ( x ) Γ ( \frac{1}{3} , i cos ^{3} ( x ) ) )}{6 3 cos ^{6} ( x )}

求解步骤

\int_{c o s (x)}^{5 x} cos (u^{3}) d u

这是非初等函数积分:

\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{u ( 3 i u ^{3} Γ ( \frac{1}{3} , - i u ^{3} ) + 3 - i u ^{3} Γ ( \frac{1}{3} , i u ^{3} ) )}{6 3 u ^{6}}_{c o s (x)}^{5 x}

带入上下限计算后得:

- \frac{5 x ( 3 i ( 5 x ) ^{3} Γ ( \frac{1}{3} , - i ( 5 x ) ^{3} ) + 3 - i ( 5 x ) ^{3} Γ ( \frac{1}{3} , i ( 5 x ) ^{3} ) )}{6 3 ( 5 x ) ^{6}} - - \frac{cos ( x ) ( 3 i cos ^{3} ( x ) Γ ( \frac{1}{3} , - i cos ^{3} ( x ) ) + 3 - i cos ^{3} ( x ) Γ ( \frac{1}{3} , i cos ^{3} ( x ) ) )}{6 3 cos ^{6} ( x )}

= - \frac{5 x ( 3 i ( 5 x ) ^{3} Γ ( \frac{1}{3} , - i ( 5 x ) ^{3} ) + 3 - i ( 5 x ) ^{3} Γ ( \frac{1}{3} , i ( 5 x ) ^{3} ) )}{6 3 ( 5 x ) ^{6}} - - \frac{cos ( x ) ( 3 i cos ^{3} ( x ) Γ ( \frac{1}{3} , - i cos ^{3} ( x ) ) + 3 - i cos ^{3} ( x ) Γ ( \frac{1}{3} , i cos ^{3} ( x ) ) )}{6 3 cos ^{6} ( x )}

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