Integrate using trigo substitution $int dx/(sqrt(x^2-4x))^3$ ?

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Answer 1 · 2018-07-14T18:45:17

$(2-x)/(4sqrt((x-4)x)+C$

At first we Substitute $u=x-2,du=dx$

and we get

$int1/(u^2-4)^(3/2)du$
now we substitue $u=2sec(s),du=2tan(s)sec(s)ds$

then we get
$1/4int cot(s)csc(s)ds=-csc(s)/4+C$
with

$s=sec(-1)(u/2)$ we get

$-1/4csc(sec^(-1)(u/2))+C$
note that

$csc(sec^(-1)(z))=1/sqrt(1-1/z^2)$ then we get

$-u/(4sqrt(u^2-4))+C$
with $u=x-2$

we get the result
$(2-x)/(4sqrt((x-4)x)+C$

Integrate using trigo substitution int dx/(sqrt(x^2-4x))^3int dx/(sqrt(x^2-4x))^3 ?