How do you integrate 1/(1-x)sqrtx^2-2x+2 ?

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Answer 1 · 2018-01-18T11:27:40

# ln|(1+sqrt(x^2-2x+2))/(1-x)|+C#.

Here is another Method to find the Integral :

Let, #(1-x)=1/t :. -dx=-1/t^2dt, or, dx=1/t^2dt#.

Also, #sqrt(x^2-2x+2)=sqrt{(1-x)^2+1}#.

#:. int1/((1-x)sqrt(x^2-2x+2))dx#,

#=int1/{1/t*sqrt(1/t^2+1)}(1/t^2)dt#,

#=int1/sqrt(t^2+1)dt#,

#=ln|t+sqrt(t^2+1)|#,

#=ln|1/(1-x)+sqrt(1/(1-x)^2+1)|#,

#=ln|(1+sqrt(x^2-2x+2))/(1-x)|+C#.