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

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

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

Solution:

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

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

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

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