What is the integral from 0 to 1 of cos(x^4) ?

The integral from 0 to 1 of cos(x^4) is -1/8 (\E_{3/4}(1i)+\E_{3/4}(-1i))

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integral from 0 to 1 of cos(x^4)

\int_{0}^{1} cos (x^{4}) d x

- \frac{1}{8} (E_{\frac{3}{4}} (1 i) + E_{\frac{3}{4}} (- 1 i))

Solution steps

\int_{0}^{1} cos (x^{4}) d x

Use the nonelementary integral:

\int cos (x^{4}) d x = - \frac{1}{8} x (E_{\frac{3}{4}} (- i x^{4}) + E_{\frac{3}{4}} (i x^{4}))

= [- \frac{1}{8} x (E_{\frac{3}{4}} (- i x^{4}) + E_{\frac{3}{4}} (i x^{4}))]_{0}^{1}

Compute the boundaries:

- \frac{1}{8} (E_{\frac{3}{4}} (1 i) + E_{\frac{3}{4}} (- 1 i))

= - \frac{1}{8} (E_{\frac{3}{4}} (1 i) + E_{\frac{3}{4}} (- 1 i))

What is the integral from 0 to 1 of cos(x^4) ?
The integral from 0 to 1 of cos(x^4) is -1/8 (\E_{3/4}(1i)+\E_{3/4}(-1i))