Question

The result of this integral is this?

`int1/xsqrt((x-2)/(x+2))dx = 2ln|x-2| - ln|x| + c`

Answer 1

`ln|t+1|-ln-1|-2arctan(t)+C` ,where `t=sqrt((x-2)/(x+2))`

Explanation:

Substituting `t=sqrt((x-2)/(x+2))` then we get

`x=-2(t^2+1)/(t^2-1)`
and

`dx=8t/((t-1)^2(t+1)^2)dt`

then we get the integral

`-1/2*int ((t^2-1)/(t^2+1))*8t^2/((t-1)^2(t+1)^2)dt`

this simplifies to

`-4int t^2/(t^4-1)dt=-4int (1/(t+1)-1/(t-1)-2/(t^2+1))dt`
with the result above.