What is the integral of this and how to get to the answer? (substitution method)

Question

#(12x)/(x^2-1)^4#

The substitution method confuses me :( I have seen the answer but don't understand it :(

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Answer 1 · 2018-05-01T15:15:22

To simplify the denominator let us substitute

#u=(x^2-1)#

Differentiating both sides with respective variables we get

#du=2x\ dx#
#=> dx=du 1/(2x)#

Integral becomes

#I=int\ (12x)/u^4 1/(2x)\ du#
#=>I=6int\ u^-4\ du#

Integrating using power rule we get

#I=6xx u^(-4+1)/(-4+1)#
#=>I=6xx u^(-3)/(-3)#
#=>I=-2/ u^(3)#

Undo substitution and add a constant of integration

#I=-2/ (x^2-1)^(3)+C#