Improper rational function: divide the polynomials first

1. A.
   $\frac{x^3}{3}-\frac{x^2}{2}+x+c$
2. B.
   $\frac{x^3}{3}+\frac{x^2}{2}+x+c$
3. C.
   $\frac{x^3}{3}-\frac{x^2}{2}-x+c$
4. D.
   $\frac{x^3}{3}+\frac{x^2}{2}-x+c$

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1. A.
   $\frac{x^2}{2}+3x-\log|x|+c$
2. B.
   $\frac{x^2}{2}+3x+2\log|x|+c$
3. C.
   $\frac{x^2}{2}-3x+2\log|x|+c$
4. D.
   $\frac{x^2}{2}-3x-\log|x|+c$

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1. A.
   $\dfrac{x^2}{2} - 2x - \dfrac{3}{2}\log|x^2 - x + 1| + \dfrac{1}{\sqrt{3}}\tan^{-1}\left(\dfrac{2x - 1}{\sqrt{3}}\right) + c$
2. B.
   $\dfrac{x^2}{2} + 2x + \dfrac{3}{2}\log|x^2 - x + 1| + \dfrac{1}{\sqrt{3}}\tan^{-1}\left(\dfrac{2x - 1}{\sqrt{3}}\right) + c$
3. C.
   $\dfrac{x^2}{2} + 2x - \dfrac{3}{2}\log|x^2 - x + 1| - \dfrac{1}{\sqrt{3}}\tan^{-1}\left(\dfrac{2x - 1}{\sqrt{3}}\right) + c$
4. D.
   $\dfrac{x^2}{2} - 2x - \dfrac{3}{2}\log|x^2 - x + 1| - \dfrac{1}{\sqrt{3}}\tan^{-1}\left(\dfrac{2x - 1}{\sqrt{3}}\right) + c$

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