x-6-x-2-x-8-x-4-1-dx-

Question Number 86824 by Ar Brandon last updated on 31/Mar/20

\int \frac{x^{6} + x^{2}}{x^{8} - x^{4} + 1} d x

Answered by TANMAY PANACEA. last updated on 31/Mar/20

\int \frac{x^{2} + \frac{1}{x^{2}}}{x^{4} - 1 + \frac{1}{x^{4}}} d x

\int \frac{x^{2} + \frac{1}{x^{2}}}{{(x^{2} + \frac{1}{x^{2}})}^{2} - 3} d x

\frac{1}{2} \int \frac{x^{2} + \frac{1}{x^{2}} + \sqrt{3} + x^{2} + \frac{1}{x^{2}} - \sqrt{3}}{(x^{2} + \frac{1}{x^{2}} + \sqrt{3}) (x^{2} + \frac{1}{x^{2}} - \sqrt{3})} d x

\frac{1}{2} \int \frac{d x}{x^{2} + \frac{1}{x^{2}} - \sqrt{3}} + \frac{1}{2} \int \frac{d x}{x^{2} + \frac{1}{x^{2}} + \sqrt{3}}

\frac{1}{4} \int \frac{1 - \frac{1}{x^{2}} + 1 + \frac{1}{x^{2}}}{x^{2} + \frac{1}{x^{2}} - \sqrt{3}} d x + \frac{1}{4} \int \frac{1 - \frac{1}{x^{2}} + 1 + \frac{1}{x^{2}}}{x^{2} + \frac{1}{x^{2}} + \sqrt{3}} d x

\frac{1}{4} \int \frac{d (x + \frac{1}{x})}{{(x + \frac{1}{x})}^{2} - (2 + \sqrt{3})} + \frac{1}{4} \int \frac{d (x - \frac{1}{x})}{{(x - \frac{1}{x})}^{2} + 2 - \sqrt{3}} + \frac{1}{4} \int \frac{d (x + \frac{1}{x})}{{(x + \frac{1}{x})}^{2} - 2 + \sqrt{3}} + \frac{1}{4} \int \frac{d (x - \frac{1}{x})}{{(x - \frac{1}{x})}^{2} + 2 + \sqrt{3}}

\frac{1}{4} \times \frac{1}{2 \sqrt{2 + \sqrt{3}}} l n (\frac{x + \frac{1}{x} - \sqrt{2 + \sqrt{3}}}{x + \frac{1}{x} + \sqrt{2 + \sqrt{3}}}) = I_{1}

\frac{1}{4} \times \frac{1}{\sqrt{2 - \sqrt{3}}} {t a n}^{- 1} (\frac{x - \frac{1}{x}}{\sqrt{2 - \sqrt{3}}}) = I_{2}

\frac{1}{4} \times \frac{1}{2 \sqrt{2 - \sqrt{3}}} l n (\frac{x + \frac{1}{x} - \sqrt{2 - \sqrt{3}}}{x + \frac{1}{x} + \sqrt{2 - \sqrt{3}}}) = I_{3}

\frac{1}{4} \times \frac{1}{\sqrt{2 + \sqrt{3}}} {t a n}^{- 1} (\frac{x - \frac{1}{x}}{\sqrt{2 + \sqrt{3}}})

p l s a d d t h e m \dots

Commented by Ar Brandon last updated on 02/Apr/20

A m a z i n g!

Commented by TANMAY PANACEA. last updated on 02/Apr/20

t h a n k y o u s i r \dots

Answered by M±th+et£s last updated on 03/Apr/20

\frac{1}{2} \int \frac{2 (x^{6} + x^{2})}{x^{8} + 2 x^{4} + 1 - 3 x^{4}} d x = \frac{1}{2} \int \frac{x^{6} + x^{6} + \sqrt{3} x^{4} - \sqrt{3} x^{4} + x^{2} + x^{2}}{{(x^{4} + 1)}^{2} - 3 x^{4}} d x

\frac{1}{2} \int \frac{x^{6} + \sqrt{3} x^{4} + x^{2} + x^{6} - \sqrt{3} x^{4} + x^{2}}{(x^{4} + 1 - \sqrt{3} x^{2}) (x^{4} + 1 + \sqrt{3} x^{2})} d x

\frac{1}{2} \int \frac{x^{2} (x^{4} + \sqrt{3} x^{2} + 1) + x^{2} (x^{4} - \sqrt{3} x^{2} + 1)}{(x^{4} + 1 - \sqrt{3} x^{2}) (x^{4} + 1 + \sqrt{3} x^{2})} d x

\frac{1}{4} \int (\frac{2 x^{2}}{(x^{4} + 1 + \sqrt{3} x^{2})} + \frac{2 x^{2}}{(x^{4} + 1 - \sqrt{3} x^{2}}) d x

\frac{1}{4} \int (\frac{x^{2} + 1 + x^{2} - 1}{(x^{4} + 1 + \sqrt{3} x^{2})} + \frac{x^{2} + 1 + x^{2} - 1}{(x^{4} + 1 - \sqrt{3} x^{2})}) d x

\frac{1}{4} \int (\frac{x^{2} + 1}{x^{4} + 1 + \sqrt{3} x^{2}} + \frac{x^{2} - 1}{x^{4} + 1 + \sqrt{3} x^{2}} + \frac{x^{2} + 1}{x^{4} + 1 - \sqrt{3} x^{2}} + \frac{x^{2} - 1}{x^{4} + 1 - \sqrt{3} x^{2}}) d x

\frac{1}{4} \int (\frac{1 + 1 / x^{2}}{x^{2} + \sqrt{3} + 1 / x^{2}} d x + \frac{1 - 1 / x^{2}}{x^{2} + \sqrt{3} + 1 / x^{2}} + \frac{1 + 1 / x^{2}}{x^{2} - \sqrt{3} + 1 / x^{2}} + \frac{1 - 1 / x^{2}}{x^{2} - \sqrt{3} + 1 / x^{2}}) d x

\frac{1}{4} \int (\frac{1 + 1 / x^{2}}{x^{2} - 2 + 1 / x^{2} - \sqrt{3} + 2} + \frac{1 - 1 / x^{2}}{x^{2} + 2 + 1 / x^{2} - 2 - \sqrt{3}} + \frac{1 + 1 / x^{2}}{x^{2} - 2 + 1 / x^{2} + 2 + \sqrt{3}} + \frac{1 - 1 / x^{2}}{x^{2} + 2 + 1 / x^{2} - 2 + \sqrt{3}})

\frac{1}{4} \int (\frac{1 + 1 / x^{2}}{{(x - 1 / x)}^{2} - (\sqrt{3} - 2)} + \frac{1 - 1 / x^{2}}{{(x + 1 / x)}^{2} - (2 + \sqrt{3})} + \frac{1 + 1 / x^{2}}{{(x - 1 / x)}^{2} + (\sqrt{3} + 2)} + \frac{1 - 1 / x^{2}}{{(x - 1 / x)}^{2} + (\sqrt{3} - 2)}) d x

\frac{1}{4 \sqrt{2 - \sqrt{3}}} {t a n h}^{- 1} \frac{1}{\sqrt{2 - \sqrt{3}}} (x - \frac{1}{x}) + \frac{1}{4 \sqrt{2 - \sqrt{3}}} {t a n h}^{- 1} \frac{1}{\sqrt{2 - \sqrt{3}}} (x + \frac{1}{x}) + \frac{1}{4 \sqrt{\sqrt{3} + 2}} {t a n}^{- 1} \frac{1}{\sqrt{\sqrt{3} + 2}} (x - \frac{1}{x}) + \frac{1}{4 \sqrt{\sqrt{3} - 2}} {t a n}^{- 1} \frac{1}{\sqrt{\sqrt{3} - 2}} (x + \frac{1}{x}) + c

p l s c h e c k