Question

If $\int \frac{x-1}{(x+1)\sqrt{x^3+x^2+x}}dx=K{\tan }^{-1}\left(f\right(x\left)\right)+C,$ then which of the following is/are true?
$($Where $K$ is fixed constant and $C$ is integration constant$)$

• A. $K=2$
• B. $f\left(x\right)=(x+\frac{1}{x}+1)$
• C. $f\left(x\right)=\sqrt{x+\frac{1}{x}+1}$
• D. $K=\frac{1}{\sqrt{2}}$

Answer 1

Answer: (A), (C)

$f\left(x\right)=\sqrt{x+\frac{1}{x}+1}$

Let, $I=\int \frac{x-1}{(x+1)\sqrt{x^3+x^2+x}}dx$
Multiplying by $(x+1)$ in numerator and denominator
$\Rightarrow I=\int \frac{x^2-1}{(x^2+2x+1)\sqrt{x^3+x^2+x}}dx=\int \frac{(1-\frac{1}{x^2})dx}{(x+\frac{1}{x}+2)\sqrt{x+\frac{1}{x}+1}}$
Put $x+\frac{1}{x}+1=p^2$
$\Rightarrow (1-\frac{1}{x^2})dx=2pdp$
$\Rightarrow I=\int \frac{2p}{(p^2+1)\left(p\right)}dp=\int \frac{2dp}{p^2+1}=2{\tan }^{-1}p=2{\tan }^{-1}{(x+\frac{1}{x}+1)}^{\frac{1}{2}}+C$
$\therefore K=2$ and $f\left(x\right)=\sqrt{x+\frac{1}{x}+1}$