Question

Let $f(x)$ be a polynomial function of degree 2 satisfying
$\int \frac{ f ( x )}{ x ^3-1} dx =\ln \left|\frac{ x ^2+ x +1}{ x -1}\right|+\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 x +1}{\sqrt{3}}\right)+ C ,$
where $C$ is indefinite integration constant.
The value of $f(1)$ is equal to

• A. 1
• B. 2
• C. -1
• D. -3

Answer 1

Answer: (D)

$\int \frac{ f ( x )}{ x ^3-1} dx =\ln \left|\frac{ x ^2+ x +1}{ x -1}\right|+\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 x +1}{\sqrt{3}}\right)+ C$
Differentiating both sides, we get
$\frac{f(x)}{x^3-1}= \frac{x-1}{x^2+x+1} \frac{(x-1)(2 x+1)-\left(x^2+x+1\right) \cdot 1}{(x-1)^2}+\frac{2}{\sqrt{3}} \cdot \frac{1}{1+\left(\frac{2 x+1}{\sqrt{3}}\right)^2} \cdot \frac{2}{\sqrt{3}} $
$ =\frac{2 x^2-x-1-x^2-x-1}{x^3-1}+\frac{4}{3} \cdot \frac{3}{3+\left(4 x^2+4 x+1\right)}=\frac{x^2-2 x-2}{x^3-1}+\frac{1}{x^2+x+1}$
$f(x)=\left(x^2-2 x-2\right)+(x-1)=x^2-x-3 \Rightarrow f(1)=-3 $