How do you integrate $\frac{5 x}{{(x - 2)}^{\frac{1}{2}}} d x$ $(5x)/ (x-2)^(1/2) dx$ ?

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Answer 1 · 2015-04-11T15:04:20

$f (x) = 5 \int \frac{x}{{(x - 2)}^{\frac{1}{2}}} d x$ $f(x)=5intx/(x-2)^(1/2)dx$

Let's $u = x - 2$ $u = x-2$

$d u = 1$ $du = 1$

$x = u + 2$ $x = u +2$

Now we have : $= 5 \int \frac{u + 2}{u^{\frac{1}{2}}} d u$ $=5int(u+2)/u^(1/2) du$

$= 5 \int \sqrt{u} d u + 10 \int \frac{1}{\sqrt{u}} d u$ $=5intsqrt(u)du+10int1/sqrt(u)du$

$= [\frac{10}{3} \sqrt{u^{3}}] + 20 [\sqrt{u}] + C$ $=[10/3sqrt(u^3)]+20[sqrt(u)]+C$

Substitute back

$\frac{10}{3} {(x - 2)}^{\frac{3}{2}} + 20 {(x - 2)}^{\frac{1}{2}} + C$ $10/3(x-2)^(3/2)+20(x-2)^(1/2)+C$

How do you integrate 5x(x−2)12dx(5x)/ (x-2)^(1/2) dx?