Question

\(\int \frac{x^{3}+x^{2}+2x+1}{x^{2}-x+1}dx=\ldots\)

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

Hint:

Divide the cubic by the quadratic, then split the remainder into a log term and an inverse tangent term.

Answer 1

The Correct Option is B

Step 1: Differentiate-check idea
The derivative of the denominator $x^{2}-x+1$ is $2x-1$, which gives the log term after division.

Step 2: Evaluate
After long division, $\int(x+2)\,dx=\frac{x^{2}}{2}+2x$, and the remainder $\frac{3x-1}{x^{2}-x+1}$ integrates to $\frac{3}{2}\log|x^{2}-x+1|+\frac{1}{\sqrt{3}}\tan^{-1}\frac{2x-1}{\sqrt{3}}$.

Step 3: Compare the options
The signs of $2x$, the log term and the inverse tangent term are all positive, which matches the second option.

Final Answer:
All terms are positive, matching the second option. \[ \boxed{\text{(B) }\dfrac{x^{2}}{2}+2x+\dfrac{3}{2}\log |x^{2}-x+1|+\dfrac{1}{\sqrt{3}}\tan ^{-1}\left(\dfrac{2x-1}{\sqrt{3}}\right)+c} \]