What is the integral from x^2 to pi of sin(θ^3) ?

The integral from x^2 to pi of sin(θ^3) is

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integral from x^2 to pi of sin(θ^3)

\int_{x^{2}}^{π} sin (θ^{3}) d θ

- \frac{1}{2} i (\frac{π Γ ( \frac{1}{3} , i π ^{3} )}{3 3 i π ^{3}} - \frac{π Γ ( \frac{1}{3} , - i π ^{3} )}{3 3 - i π ^{3}}) - - \frac{1}{2} i \frac{x ^{2} Γ ( \frac{1}{3} , i ( x ^{2} ) ^{3} )}{3 3 i ( x ^{2} ) ^{3}} - \frac{x ^{2} Γ ( \frac{1}{3} , - i ( x ^{2} ) ^{3} )}{3 3 - i ( x ^{2} ) ^{3}}

Solution steps

\int_{x^{2}}^{π} sin (θ^{3}) d θ

Use the nonelementary integral:

\int sin (x^{3}) d x = - \frac{1}{2} i (\frac{x Γ ( \frac{1}{3} , i x ^{3} )}{3 3 i x ^{3}} - \frac{x Γ ( \frac{1}{3} , - i x ^{3} )}{3 3 - i x ^{3}})

= [- \frac{1}{2} i (\frac{θ Γ ( \frac{1}{3} , i θ ^{3} )}{3 3 i θ ^{3}} - \frac{θ Γ ( \frac{1}{3} , - i θ ^{3} )}{3 3 - i θ ^{3}})]_{x^{2}}^{π}

Compute the boundaries:

- \frac{1}{2} i (\frac{π Γ ( \frac{1}{3} , i π ^{3} )}{3 3 i π ^{3}} - \frac{π Γ ( \frac{1}{3} , - i π ^{3} )}{3 3 - i π ^{3}}) - - \frac{1}{2} i \frac{x ^{2} Γ ( \frac{1}{3} , i ( x ^{2} ) ^{3} )}{3 3 i ( x ^{2} ) ^{3}} - \frac{x ^{2} Γ ( \frac{1}{3} , - i ( x ^{2} ) ^{3} )}{3 3 - i ( x ^{2} ) ^{3}}

= - \frac{1}{2} i (\frac{π Γ ( \frac{1}{3} , i π ^{3} )}{3 3 i π ^{3}} - \frac{π Γ ( \frac{1}{3} , - i π ^{3} )}{3 3 - i π ^{3}}) - - \frac{1}{2} i \frac{x ^{2} Γ ( \frac{1}{3} , i ( x ^{2} ) ^{3} )}{3 3 i ( x ^{2} ) ^{3}} - \frac{x ^{2} Γ ( \frac{1}{3} , - i ( x ^{2} ) ^{3} )}{3 3 - i ( x ^{2} ) ^{3}}

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What is the integral from x^2 to pi of sin(θ^3) ?
The integral from x^2 to pi of sin(θ^3) is