Use the substitution u=x^2-2 to find the integral of (6x^3+4x)/sqrt( x^2-2)

First use the substitution to find du/dx which is 2x. From this we now know that dx= du/2x (just re-arranging.) Substituting that into the integral we now get (6x³+4x)/ (sqrt (u) x 2x) du. Cancelling out the 2x we now have (3x²+2)/ sqrt (u) du. This is equal to (3u+8)/ sqrt(u) du using the original definition of u. This is equal to 3u^0.5+ 8u^-0.5 du. Integrating this we get 2u^1.5 + 16u^0.5+ c and to get the final answer we just substitute for u.