Question

Let $f(x)$ be a polynomial function of degree 2 satisfying
$\int \frac{f(x)}{x^3-1}dx=\ln \left|\frac{x^2+x+1}{x-1}\right|+\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+C,$
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)

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