If ∫f(x)x2−x+1dx=32log⁡x2−x+1+13tan−1⁡2x−13+C  then f(x) is equal to

# If $\int \frac{f\left(x\right)}{{x}^{2}-x+1}\mathrm{d}x=\frac{3}{2}\mathrm{log}\left({x}^{2}-x+1\right)+\frac{1}{\sqrt{3}}{\mathrm{tan}}^{-1}\frac{2x-1}{\sqrt{3}}+C$  then $f\left(x\right)$ is equal to

1. A

$3x$

2. B

$3x-4$

3. C

$3x-1$

4. D

$\frac{1}{1+{x}^{2}}$

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### Solution:

Differentiating both the sides w.r.t. x , we get

$\frac{f\left(x\right)}{{x}^{2}-x+1}=\frac{3}{2}\frac{2x-1}{{x}^{2}-x+1}+\frac{1}{\sqrt{3}}\frac{1}{1+\left(\frac{2x-1}{\sqrt{3}}{\right)}^{2}}\left(\frac{2}{\sqrt{3}}\right)$

$\begin{array}{l}=\frac{3}{2}\frac{2x-1}{{x}^{2}-x+1}+\frac{1}{2\left({x}^{2}-x+1\right)}\\ ⇒f\left(x\right)=3x-1\end{array}$

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