The result of this integral is this?

Question

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

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Answer 1 · 2018-06-21T18:24:44

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

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.