Integrate (x+3)/(x(x-3)) with respect to x

The easiest way to solve this is to split the fraction into partials.Using partial fractions, we get A/(x-3)+B/x=x+3/(x(x-3)) implies Ax+B(x-3)=x+3We want to find the values of A and B that solve this, so we want to eliminate A and solve for B, then eliminate B and solve for A.Setting x=0 eliminates A, so B(-3)=3 implies that B=-1Setting x=3 eliminates B, so 3A = 6 implies A=2Thus we have 2/(x-3)-1/xWe can integrate this fine now.The integral of 2/(x-3) is going to be 2ln|x-3|, as the numerator 2 times the derivative of the denominator. Likewise, -1/x integrates to -ln|x|. So the integral is 2ln|x-3|-ln|x|+c