dx-x-4-x-2-a-2-

Question Number 146147 by iloveisrael last updated on 11/Jul/21

Υ = \int \frac{dx}{x^{4} \sqrt{x^{2} - a^{2}}} = ?

Answered by EDWIN88 last updated on 11/Jul/21

S o l v e \int \frac{d x}{x^{4} \sqrt{x^{2} - a^{2}}}

Υ = \int \frac{d x}{x^{2} . x^{3} \sqrt{1 - a^{2} x^{- 2}}}

l e t u = 1 - a^{2} x^{- 2} \to \frac{d u}{2 a^{2}} = \frac{d x}{x^{3}}

a n d \frac{a^{2}}{x^{2}} = 1 - u \Rightarrow \frac{1}{x^{2}} = \frac{1 - u}{a^{2}}

Υ = \int \frac{(1 - u) d u}{2 a^{4} . \sqrt{u}} = \frac{1}{2 a^{4}} \int (u^{- 1 / 2} - u^{1 / 2}) d u

Υ = \frac{1}{2 a^{4}} (2 \sqrt{u} - \frac{2}{3} u \sqrt{u}) + c

Υ = \frac{\sqrt{u}}{a^{4}} (1 - \frac{1}{3} u) + c

Υ = \frac{\sqrt{a^{2} - x^{2}}}{a^{4} x} (1 - \frac{x^{2} - a^{2}}{x^{2}}) + c

Υ = \frac{\sqrt{a^{2} - x^{2}}}{a^{2} x^{3}} + c

Answered by mathmax by abdo last updated on 11/Jul/21

Υ =_{x = \frac{a}{sint}} \int \frac{\sin^{4} t}{a^{4} \sqrt{\frac{a^{2}}{\sin^{2} t} - a^{2}}} (- \frac{acost}{\sin^{2} t}) dt

= \frac{1}{a^{4}} \int \frac{\sin^{2} t . cost}{cost} sint dt = \frac{1}{a^{4}} \int \sin^{3} t dt

\sin^{3} t = sint (\frac{1 - \cos (2 t)}{2}) = \frac{1}{2} (sint - sint \cos (2 t))

sint \cos (2 t) = \cos (\frac{π}{2} - t) \cos (2 t) = \frac{1}{2} {\cos (\frac{π}{2} + t) + \cos (\frac{π}{2} - 3 t))

= \frac{1}{2} {- sint + \sin (3 t)} \Rightarrow \sin^{3} t = \frac{1}{2} sint - \frac{1}{4} {- sint + \sin (3 t)}

= \frac{3}{4} sint - \frac{1}{4} \sin (3 t) \Rightarrow Υ = \frac{1}{a^{4}} \int (\frac{3}{4} sint - \frac{1}{4} \sin (3 t)) dt

= - \frac{3}{{4 a}^{4}} cost + \frac{1}{{12 a}^{4}} \cos (3 t) + K

sint = \frac{a}{x} \Rightarrow cost = \sqrt{1 - \frac{a^{2}}{x^{2}}}

\cos (3 t) = cost . \cos 2 t - sint \sin (2 t)

= cost ({2 \cos}^{2} t - 1) - {2 \sin}^{2} t cost

= \sqrt{1 - \frac{a^{2}}{x^{2}}} (2 (1 - \frac{a^{2}}{x^{2}}) - 1) - 2 \frac{a^{2}}{x^{2}} \sqrt{1 - \frac{a^{2}}{x^{2}}} \Rightarrow

Ψ = - \frac{3}{{4 a}^{4}} \sqrt{1 - \frac{a^{2}}{x^{2}}} + \frac{1}{{12 a}^{4}} \sqrt{1 - \frac{a^{2}}{x^{2}}} (1 - \frac{{4 a}^{2}}{x^{2}}) + K