sin-x-4-dx-

Question Number 77716 by aliesam last updated on 09/Jan/20

\int s i n (x^{4}) d x

Commented by MJS last updated on 09/Jan/20

this can be solved using the incomplete

gamma function . I^{'} ll post it later

Commented by aliesam last updated on 09/Jan/20

t h a n k s s i r

Answered by MJS last updated on 09/Jan/20

\int \sin x^{4} d x = - \frac{i}{2} \int (e^{i x^{4}} - e^{- i x^{4}}) d x =

[t = x \sqrt[4]{- i} = \frac{x}{2} (\sqrt{2 + \sqrt{2}} - i \sqrt{2 - \sqrt{2}}) \to d x = \frac{d t}{2} (\sqrt{2 + \sqrt{2}} + i \sqrt{2 - \sqrt{2}})]

= \frac{1}{4} (\sqrt{2 - \sqrt{2}} - i \sqrt{2 + \sqrt{2}}) \int (e^{- t^{4}} + e^{t^{4}}) d t

\int e^{- t^{4}} d t = - \frac{\sqrt{2}}{8} (1 - i) Γ (\frac{1}{4}, - t^{4})

\int e^{t^{4}} d t = - \frac{1}{4} Γ (\frac{1}{4}, t^{4})

\dots

Commented by aliesam last updated on 09/Jan/20

p e r f e c t s i r t h a n k s

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