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

Problem 4.171 · medium

1. \[ \int \frac{x^{2} + 1}{x \left(x - 1\right)}\, dx \]
   integral algebraStart with the integral of the given function. Rewrite the numerator to facilitate long division.✓ Proved
2. \[ = \int \frac{x \left(x - 1\right) + x + 1}{x \left(x - 1\right)}\, dx \]
   algebraGroup terms to isolate the polynomial part.✓ Proved
3. \[ = \int \left(1 + \frac{x + 1}{x \left(x - 1\right)}\right)\, dx \]
   long-divisionPerform long division on the rational part.✓ Proved
4. \[ = \int 1\, dx + \int \frac{x + 1}{x \left(x - 1\right)}\, dx \]
   linearity algebraSplit the integral into two parts. Rewrite the numerator of the second term.✓ Proved
5. \[ = \int 1\, dx + \int \left(\frac{1}{x} + \frac{2}{x \left(x - 1\right)}\right)\, dx \]
   algebraDistribute the denominator.✓ Proved
6. \[ = \int 1\, dx + \int \frac{1}{x}\, dx + \int \frac{2}{x \left(x - 1\right)}\, dx \]
   linearity algebra simplifySplit the integral into three parts. Rewrite the denominator for partial fraction decomposition. Simplify the expression back to a standard form.✓ Proved
7. \[ = \int 1\, dx + \int \frac{1}{x}\, dx + \int \left(\frac{2}{x - 1} - \frac{2}{x}\right)\, dx \]
   partial-fractionsApply partial fraction decomposition to the last term.✓ Proved
8. \[ = \int 1\, dx + \int \frac{1}{x}\, dx - \int \frac{2}{x}\, dx + \int \frac{2}{x - 1}\, dx \]
   linearityDistribute the integral across the subtraction.✓ Proved
9. \[ = x - \ln{\left(x \right)} + 2 \ln{\left(x - 1 \right)} \]
   antiderivative simplifyIntegrate each term separately. Combine the logarithmic terms.✓ Proved