`int (8x) / (x^3+x^2-x-1) dx` Use partial fractions to find the indefinite integral

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Now let's form the partial fraction template,


Multiply the equation by the denominator,





Comparing the coefficients of the like terms,

`A+B=0`      -----------------(1)

`2A+C=8`   -----------------(2)

`A-B-C=0`  ---------------(3)

From equation 1,


Substitute B in equation 3,


`2A-C=0`      ---------------(4)

Now add equations 2 and 4,





Plug in the value of A in equation 4,



Plug in the values of A, B and C in the partial fraction template,



Apply the sum rule,


Take the constant out,


Now let's evaluate the above three integrals separately,


Apply integral substitution:`u=x-1`



use the common integral `int1/xdx=ln|x|`


Substitute back `u=x-1`


Now let's evaluate second integral,


Apply integral substitution: `u=x+1`




Substitute back `u=x+1`


Now evaluate the third integral,


apply integral substitution: `u=(x+1)`




apply power rule,



Substitute back `u=x+1`



Simplify and add a constant C to the solution,



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