x-3-x-6-1-dx-

Question Number 41084 by Tawa1 last updated on 01/Aug/18

\int \frac{x^{3}}{x^{6} + 1} dx

Answered by tanmay.chaudhury50@gmail.com last updated on 01/Aug/18

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

t = x^{2} d t = 2 x d x

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

\frac{t}{(t + 1) (t^{2} - t + 1)} = \frac{p}{t + 1} + \frac{q t + r}{t^{2} - t + 1}

t = {p t}^{2} - p t + p + {q t}^{2} + r t + q t + r

t = t^{2} (p + q) + t (- p + q + r) + p + r

p + q = 0

- p + q + r = 1

p + r = 0

r + r + r = 1 r = \frac{1}{3} p = - \frac{1}{3} q = \frac{1}{3}

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

= \frac{- 1}{3} \int \frac{d t}{t + 1} + \frac{1}{3} \int \frac{t + 1}{t^{2} - t + 1} d t

. \frac{- 1}{3} \int \frac{d t}{t + 1} + \frac{1}{6} \int \frac{2 t - 1 + 3}{t^{2} - t + 1} d t

\frac{- 1}{3} \int \frac{d t}{t + 1} + \frac{1}{6} \int \frac{2 t - 1}{t^{2} - t + 1} d t + \frac{3}{6} \int \frac{d t}{t^{2} - 2 t . \frac{1}{2} + \frac{1}{4} + 1 - \frac{1}{4}}

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

\frac{- 1}{3} l n (t + 1) + \frac{1}{6} l n (t^{2} - t + 1) + \frac{3}{6} \times \frac{2}{\sqrt{3}} {t a n}^{- 1} (\frac{t - \frac{1}{2}}{\frac{\sqrt{3}}{2}})

n o w p u t x^{2} i n p l a c e o f t

\frac{- 1}{3} l n (x^{2} + 1) + \frac{1}{6} l n (x^{4} - x^{2} + 1) + \frac{3}{6} \times \frac{2}{\sqrt{3}} {t a n}^{- 2} (\frac{x^{2} - \frac{1}{2}}{\frac{\sqrt{3}}{2}}

Commented by Tawa1 last updated on 01/Aug/18

Wow . God bless you sir

Commented by tanmay.chaudhury50@gmail.com last updated on 01/Aug/18

g o o d n i g h t \dots