x-x-3-1-x-2-dx-

Question Number 125313 by john_santu last updated on 10/Dec/20

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

Answered by john_santu last updated on 10/Dec/20

l e t \sqrt{1 - x^{2}} = w \Rightarrow x^{2} = 1 - w^{2}

x d x = - w d w

β (x) = \int \frac{(1 - w^{2}) (- w d w)}{w}

= \int (w^{2} - 1) d w = \frac{1}{3} w^{3} - w + c

= \frac{1}{3} w (w^{2} - 3) + c

= \frac{\sqrt{1 - x^{2}}}{3} (1 - x^{2} - 3) + c

= \frac{\sqrt{1 - x^{2}}}{3} (- 2 - x^{2}) + c

Answered by liberty last updated on 10/Dec/20

w e s u b s t i t u t e x^{2} = r t h e n x = r^{\frac{1}{2}}

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

= \frac{1}{2} \int r {(1 - r)}^{- \frac{1}{2}} d r

l e t {(1 - r)}^{\frac{1}{2}} = h, r = 1 - h^{2}; d r = - 2 h d h

β (x) = \frac{1}{2} \int (1 - h^{2}) h^{- 1} (- 2 h) d h

= \frac{1}{3} h^{3} - h + c = \frac{1}{3} h (h^{2} - 3) + c

= \frac{\sqrt{1 - r}}{3} (- r - 2) + c = \frac{\sqrt{1 - x^{2}}}{3} (- x^{2} - 2) + c

Answered by Ñï= last updated on 10/Dec/20

Commented by Ñï= last updated on 10/Dec/20

x = i t a n θ

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