Answer is: option2
\( x^3 + \frac{5x^2}{2} + 5x + 11 \ln |x - 2| + C \)Solution:
Using long division method:
\[ \frac{3x^3 - x^2 - 5x + 1}{x - 2} = 3x^2 + 5x + 5 + \frac{11}{x - 2} \]
Now integrate term by term:
\[ \int \left( 3x^2 + 5x + 5 + \frac{11}{x - 2} \right) dx = \int 3x^2 \, dx + \int 5x \, dx + \int 5 \, dx + \int \frac{11}{x - 2} \, dx \]
\[ \int \frac{3x^3 - x^2 - 5x + 1}{x - 2} \, dx = x^3 + \frac{5x^2}{2} + 5x + 11 \ln |x - 2| + C \]
Hence option B.