1-x-4-1-x-4-3-2-dx-A-A-B-Find-B-Assume-integration-of-constant-0-

Question Number 36801 by rahul 19 last updated on 05/Jun/18

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

\int A = B

Find B ?

Assume integration of constant = 0 .

Answered by MJS last updated on 05/Jun/18

A = \frac{x}{\sqrt{1 - x^{4}}}

B = \frac{1}{2} \arcsin x^{2}

A :

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

looks like {it}^{'} s \frac{p (x)}{\sqrt{1 - x^{4}}} with p (x) is a polynome

{let}^{'} s try

\frac{d}{d x} (\frac{p (x)}{\sqrt{1 - x^{4}}}) = \frac{d}{d x} (p (x) {(1 - x^{4})}^{- \frac{1}{2}}) =

= p^{'} (x) {(1 - x^{4})}^{- \frac{1}{2}} + p (x) (2 x^{3} {(1 - x^{4})}^{- \frac{3}{2}}) =

= \frac{p^{'} (x) (1 - x^{4}) + 2 p (x) x^{3}}{{(1 - x^{4})}^{\frac{3}{2}}}

\Rightarrow p^{'} (x) (1 - x^{4}) + 2 p (x) x^{3} = 1 + x^{4}

{let}^{'} s try p (x) = a x + b; p^{'} (x) = a

a + 2 {b x}^{3} + {a x}^{4} = 1 + x^{4}

a = 1; b = 0

p (x) = x

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

B :

\int \frac{x}{\sqrt{1 - x^{4}}} d x =

[t = x^{2} \to d x = \frac{d t}{2 x}]

= \frac{1}{2} \int \frac{d t}{\sqrt{1 - t^{2}}} = \frac{1}{2} \arcsin t =

= \frac{1}{2} \arcsin x^{2}

Answered by tanmay.chaudhury50@gmail.com last updated on 06/Jun/18

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

x^{2} = \frac{1}{t} x = t^{\frac{- 1}{2}} d x = \frac{- 1}{2} \times \frac{1}{t^{\frac{3}{2}}} d t

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

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

= \frac{- 1}{2} \int \frac{t^{2} + 1}{t^{2} - 1} \times \frac{1}{\sqrt{t}} \times \frac{1}{\sqrt{t} (\sqrt{t - \frac{1}{t}}} d t

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

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

= \frac{- 1}{2} \int \frac{d k}{k^{\frac{3}{2}}} w h e n (k = t - \frac{1}{t})

= \frac{- 1}{2} \times \frac{k^{\frac{- 1}{2}}}{\frac{- 1}{2}} + c

= \frac{1}{\sqrt{k}} + c

= \frac{1}{\sqrt{t - \frac{1}{t}}} + c

= \frac{1}{\sqrt{\frac{1}{x^{2}} - x^{2}}} + c

s o A = \frac{1}{\sqrt{\frac{1}{x^{2}} - x^{2}}}

\int A d x = B

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

y = x^{2}

d y = 2 x d x

\int \frac{d y}{2 \times \sqrt{1 - y^{2}}}

= \frac{1}{2} {s i n}^{- 1} (y)

= \frac{1}{2} {s i n}^{- 1} (x^{2}) i s B