Solve this ∫_ (((x−yz))/((x^2 +y^2 −2xyz)^(3/2) ))dz


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Question Number 79607 by sou99 last updated on 26/Jan/20

$S o l v e t h i s$ $\int_{} \frac{(x - y z)}{{(x^{2} + y^{2} - 2 x y z)}^{3 / 2}} d z$
Commented by MJS last updated on 26/Jan/20

$\int \frac{a + b t}{{(c + d t)}^{3 / 2}} d t =$ $[u = c + d t \to d t = \frac{d u}{d}]$ $= \frac{b}{d^{2}} \int \frac{d u}{u^{1 / 2}} + \frac{a d - b c}{d^{2}} \int \frac{d u}{u^{3 / 2}} =$ $= \frac{2 b}{d^{2}} u^{1 / 2} - \frac{2 (a d - b c)}{d^{2}} u^{- 1 / 2} =$ $= \frac{2 (b u - a d + b c)}{d^{2} \sqrt{u}} = \frac{2 (b d t - a d + 2 b c)}{d^{2} \sqrt{c + d t}} =$ $now put$ $a = x, b = - y, c = x^{2} + y^{2}, d = - 2 x y, t = z$ $= \frac{(x z - y)}{x^{2} \sqrt{x^{2} + y^{2} - 2 x y z}} + C$
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