Integral of \( \displaystyle \frac{\left(x - 1\right)^{2} + 1}{\left(x - 2\right) \left(x - 1\right)} \)

Problem 4.227 · medium

1. \[ \int \frac{\left(x - 1\right)^{2} + 1}{\left(x - 2\right) \left(x - 1\right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \frac{x^{2} - 2 x + 2}{\left(x - 2\right) \left(x - 1\right)}\, dx \]
   algebraExpand the numerator.✓ Proved
3. \[ = \int \frac{x + \left(x - 2\right) \left(x - 1\right)}{\left(x - 2\right) \left(x - 1\right)}\, dx \]
   algebraRewrite the numerator to facilitate division.✓ Proved
4. \[ = \int \left(\frac{x}{\left(x - 2\right) \left(x - 1\right)} + 1\right)\, dx \]
   long-division algebraPerform polynomial division. Split the numerator of the remainder term.✓ Proved
5. \[ = \int \left(1 + \frac{1}{x - 1} + \frac{2}{\left(x - 2\right) \left(x - 1\right)}\right)\, dx \]
   algebraSimplify the fraction by canceling (x - 2).✓ Proved
6. \[ = \int \left(1 - \frac{1}{x - 1} + \frac{2}{x - 2}\right)\, dx \]
   partial-fractions simplifyApply partial fraction decomposition to the last term. Combine like terms.✓ Proved
7. \[ = \int 1\, dx + \int \frac{2}{x - 2}\, dx - \int \frac{1}{x - 1}\, dx \]
   linearityDistribute the integral across the terms.✓ Proved
8. \[ = x + 2 \ln{\left(x - 2 \right)} - \ln{\left(x - 1 \right)} \]
   antiderivativeEvaluate the integrals.✓ Proved