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

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The given integrand is a improper rational function, as the degree of the numerator is more than the degree of the denominator. To apply the method of partial fractions,first we have to do a division with remainder.


Since the polynomials do not completely divide, we have to continue partial fractions on the remainder.

We need to factor the denominator,




Now let's create the partial fraction template,


Multiply the above equation by denominator,




Equating the coefficients of the like terms,

`A+B=2`    ------------------------(1)

`2A-B=1`  -------------------------(2)``

Now we have to solve the above two linear equations to get A and B,

Add the equations 1 and 2,




Plug in the value of A in equation 1,



Now plug in the values of A and B in the partial fraction template,


Now we can evaluate the integral as,


Apply the sum rule,


For the first and second integral apply the power rule and for the third and fourth integral use the common integral:`int1/xdx=ln|x|`


Add a constant C to the solution,


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