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Solve the following.
Evaluate: ∫ (x³ - x² + x - 1)/(x - 1) dx
Evaluate: $\int \frac{x^3 - x^2 + x - 1}{x - 1} \, dx$
Answer
Answer
Integral: $\int \left(\frac{x^3 - x^2 + x - 1}{x - 1}\right) dx$
$x^3 - x^2 + x - 1 = x^2(x-1) + 1(x-1) = (x-1)(x^2+1)$.
$\int \frac{(x-1)(x^2+1)}{x-1}\,dx = \int (x^2 + 1)\,dx$.
Hence the integral $= \frac{x^3}{3} + x + C$.
Final answer: $\frac{x^3}{3} + x + C$