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To integrate this rational function, we can perform polynomial long division or simply manipulate the numerator.Method 1: Manipulating the numeratorWe want to express the numerator (2x) in terms of the denominator (x-1).2x = 2(x-1) + 2Now, substitute this back into the integral:∫ [2(x-1) + 2] / (x-1) dx= ∫ [ (2(x-1))/(x-1) + 2/(x-1) ] dx= ∫ [2 + 2/(x-1)] dxNow, integrate each term:= ∫2 dx + ∫[2/(x-1)] dx= 2x + 2ln|x-1| + CMethod 2: Polynomial Long DivisionDividing 2x by (x-1):(2x) / (x-1) = 2 with a remainder of 2. (Since 2(x-1) = 2x - 2, then 2x - (2x-2) = 2)So, (2x) / (x-1) = 2 + 2/(x-1).Integrating this gives the same result.Both methods lead to 2x + 2ln|x-1| + C.