Question

\[\int\frac{x^3}{x - 2} dx\]

Answer 1

\[\int\left( \frac{x^3}{x - 2} \right)dx\]
\[ = \int\left( \frac{x^3 - 8 + 8}{x - 2} \right)dx\]
\[ = \int\left[ \frac{\left( x^3 - 2^3 \right)}{\left( x - 2 \right)} + \frac{8}{x - 2} \right]dx\]
\[ = \int\left[ \frac{\left( x - 2 \right)\left( x^2 + 2x + 4 \right)}{\left( x - 2 \right)} + \frac{8}{x - 2} \right]dx\]
\[ = \int\left( x^2 + 2x + 4 \right)dx + 8\int\frac{dx}{x - 2}\]
\[ = \frac{x^3}{3} + \frac{2 x^2}{2} + 4x + \text{8 ln} \left| x - 2 \right| + C\]
\[ = \frac{x^3}{3} + x^2 + 4x + \text{8 ln }\left| x - 2 \right| + C\]