Evaluate the following integral: (x^4 - x^2 +2)/(x^2(x-1)) dx

This type of question appears over-complicated with limited options, but one must not fear! At first, the numerator seems to be of a higher degree than the denominator (x^4 compared to x^2), but from the first two terms of our numerator we can factorise out a common x^2 from (x^4-x^2) to give x^2(x^2-1) and thus factorise by the difference of two squares hence bringing our numerator to: x^2(x-1)(x+1) +2. By this, we can cancel of x^2(x-1) from both numerator and denominator leaving: x + 1 + 2/(x^2(x-1)) to be integrated. Since we have three terms separated by addition, we can separate our integral in the following 3 separate integrals which we will add together at the end: (1) integrate x ; (2) integrate 1 ; (3) integrate 2/(x^2(x-1)). Beginning with (3), once again we can turn a complicated format into something simpler, this time we will use partial fractions. To use partial fractions, you must first distinguish how many terms lie in the denominator. From our denominator, x^2(x-1) we may initially believe that we only have 2 terms, but upon reflection of the rules for partial fractions, we have 3. This is since, under patricidal fractions we must have only linear factors in the denominator, thus from x^2 we derive two denominators: x and x^2. So in total we have the three denominators of x, x^2 and (x-1). Using partials fractions we can form the equality A/x + B/x^2 + C/(x-1) = 2/(x^2(x-1)). Simplifying both sides gives Ax^2 -Ax +Bx -B +Cx^2 = 2. By solving simultaneous equations, we find A=B= -2 and C=2. We can now create a simpler integral for 2/(x^2(x-1)), that being -2/x -2/x^2 +2/(x-1). Once we include integral (1) and (2), we come to the final simplified set of terms before integration: -2/x -2/x^2 +2/(x-1) +x +1. Using the principals of the natural log (ln) where the integral of 1/x = ln(x), we can integrate: -2/x = -2ln(x), while the integral of 2/(x-1) = 2ln(x-1). The integral of -2/x^2 =2/x because the negative exponent cancels out the minus sign while the integral of x+1 = (x^2)/ 2 +x. Finally bringing it all together our final answer is: -2lnx + 2/x +2ln(x-1) +(x^2)/2 +x + C. The C stands for constant which is to always be included in the answered derived from integration.