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

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Let's use partial fraction decomposition on the integrand,




Now form the partial fractions using the denominator,


Multiply equation by the denominator `(3x+1)(x-1)`




comparing the coefficients of the like terms,

`A+3B=-1`   ----------------(1)

`-A+B=3`      ----------------(2)

Now let's solve the above equations to get A and B,

Add the equations 1 and 2,





Plug in the value of B in equation 1,





Plug in the value of A and B in the partial fraction template,



So, `int(3-x)/(3x^2-2x-1)dx=int(-5/(2(3x+1))+1/(2(x-1)))dx`

Apply the sum rule,


Take the constant out,


Now let's evaluate both the above integrals separately,


Apply integral substitution:`u=3x+1`



Take the constant out,


Use the common integral:`int1/xdx=ln|x|`


Substitute back `u=3x+1`


Now evaluate the second integral.


Apply integral substitution: `u=x-1`



Use the common integral:`int1/xdx=ln|x|`


Substitute back `u=x-1`



Simplify and add a constant C to the solution,



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