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Question
Evaluate: \[\int\frac{x^3 - x^2 + x - 1}{x - 1} \text{ dx }\]
Solution
\[\int\left( \frac{x^3 - x^2 + x - 1}{x - 1} \right) dx\]
\[ = \int\left( \frac{x^2 \left( x - 1 \right) + 1\left( x - 1 \right)}{\left( x - 1 \right)} \right)dx\]
\[ = \int\frac{\left( x^2 + 1 \right) \left( x - 1 \right)}{\left( x - 1 \right)}dx\]
\[ = \int\left( x^2 + 1 \right) dx\]
\[ = \frac{x^3}{3} + x + C\]
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