Calculus of a Single Variable, Chapter 5, 5.8, Section 5.8, Problem 88

The given problem (dy)/(dx) =1/((x-1)sqrt(-4x^2+8x+1)) is in form of a first order ordinary differential equation. To evaluate this, we may follow the variable separable differential equation: N(y) dy= M(x)dx .
dy=1/((x-1)sqrt(-4x^2+8x+1)) dx
Apply direct integration on both sides:
int dy=int 1/((x-1)sqrt(-4x^2+8x+1)) dx
For the left side, we apply basic integration property: int (dy)=y.
For the right side, we apply several substitutions to simplify it.
Let u =(x-1) then x=u+1 and du=dx . The integral becomes:
int 1/((u)sqrt(-4x^2+8x+1)) dx =int 1/(usqrt(-4(u+1)^2+8(u+1)+1)) du
=int 1/(usqrt(-4(u^2+2u+1)+8u+8+1)) du
=int 1/(usqrt(-4u^2-8u-4+8u+8+1)) du
=int 1/(usqrt(-4u^2+5)) du
Let v = u^2 then dv = 2u du or (dv)/(2u)=du . The integral becomes:
int 1/(usqrt(-4u^2+5)) du=int 1/(usqrt(-4v+5)) *(dv)/(2u)
=int (dv)/(2u^2sqrt(-4v+5))
=int (dv)/(2vsqrt(-4v+5))
Apply the basic integration property: int c*f(x)dx= c int f(x) dx .
int (dv)/(2vsqrt(-4v+5)) =(1/2)int (dv)/(vsqrt(-4v+5))
Let w= sqrt(-4v+5) then v= (5-w^2)/4 and dw=-2/sqrt(-4v+5)dv or
(dw)/(-2)=1/sqrt(-4v+5)dv
The integral becomes:
(1/2)int (dv)/(vsqrt(-4v+5)) =(1/2)int 1/v*(dv)/sqrt(-4v+5)
=(1/2)int 1/((5-w^2)/4)*(dw)/(-2)
=(1/2)int 1*4/(5-w^2)*(dw)/(-2)
=(1/2)int -2/(5-w^2)dw
=(1/2)*-2 int 1/(5-w^2)dw
=(-1) int 1/(5-w^2)dw
Apply basic integration formula for inverse hyperbolic tangent function:
int (du)/(a^2-u^2)=(1/a)arctanh(u/a)+C
Then, with corresponding values as: a^2=5 and u^2=u^2 , we get: a=sqrt(5) and u=w
(-1) int 1/(5-w^2)dw = -1/sqrt(5) arctanh(w/sqrt(5))+C
Recall w=sqrt(-4v+5) and v=u^2 then w =sqrt(-4u^2+5).
Plug-in u=(x-1) on w =sqrt(-4u^2+5) , we get:
w =sqrt(-4(x-1)^2+5)
w=sqrt(-4(x^2-2x+1)+5)
w=sqrt(-4x^2+8x-4+5)
w=sqrt(-4x^2+8x+1)

Plug-in w=sqrt(-4x^2+8x+1) on -1/sqrt(5) arctanh(w/sqrt(5))+C , we get:
int 1/((x-1)sqrt(-4x^2+8x+1)) dx=1/sqrt(5)arctanh(sqrt(-4x^2+8x+1)/sqrt(5))+C
=-1/sqrt(5) arctanh(sqrt(-4x^2+8x+1)/5)+C
Combining the results from both sides, we get the general solution of the differential equation as:
y=-1/sqrt(5) arctanh(sqrt(-4x^2+8x+1)/5)+C