dx-4x-x-2-3-

Question Number 119784 by bemath last updated on 27/Oct/20

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

Answered by bobhans last updated on 27/Oct/20

\int \frac{d x}{{(4 - {(2 - x)}^{2})}^{\frac{3}{2}}}; [l e t 2 - x = 2 \sin t]

d x = - 2 \cos t d t]

\int \frac{- 2 \cos t d t}{{(4 - 4 \sin^{2} t)}^{\frac{3}{2}}} = \int \frac{- \cos t d t}{4 \cos^{3} t}

= - \frac{1}{4} \int \frac{1}{\cos^{2} t} d t = - \frac{1}{4} \tan t + c

= - \frac{2 - x}{4 \sqrt{4 x - x^{2}}} + c = \frac{x - 2}{4 \sqrt{4 x - x^{2}}} + c

Answered by MJS_new last updated on 28/Oct/20

\int \frac{d x}{x^{3 / 2} {(4 - x)}^{3 / 2}} =

[t = \frac{\sqrt{x}}{\sqrt{4 - x}} \to d x = \frac{\sqrt{x {(4 - x)}^{3}}}{2} d t]

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

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