Find ∫((x^2−2)(x^2+2)/x^2) dx, x≠0

Before we start any integration steps it would be wise to try and simplify the fraction that we have been given. We want to get rid of that denominator so we will start by expanding out the numerator. This comes to (x^4) + (2x^2) - (2x^2) - 4, and hence the 2x^2 cancel; leaving us with x^4 - 4. We can then break down the large fraction of (x^4 - 4)/(x^2) into two smaller fractions with a common denominator. We then have x^4/x^2 – 4/x^2. Simplifying this gives x^2 – 4x^-2 as when there is a power on the bottom of the fraction, the sign of that power is inverted.We can now start the integration process. We can apply integration rules to this new polynomial, giving us x^3/3 – ((4x^-1)/-1) + c => (1/3)x^3 + (4/x) + c (never forget the constant of integration when performing indefinite integrals!). We can leave our answer in this form and for most cases this would be fine but if wanted it can be simplified down to ((x^4 + 12)/(3x)) + c.