Question

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

Answer 1

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

Explanation:

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`