Question

If the integral $Ι=\displaystyle \int \frac{d x}{x^{10} + x}=$ $\lambda ln\left(\frac{x^{9}}{1 + x^{\mu }}\right)+C,$ (where, $C$ is the constant of integration) then the value of $\frac{1}{\lambda }+\mu $ is equal to

• A. $81$
• B. $\frac{82}{9}$
• C. $18$
• D. $8$

Answer 1

Answer: (C)

$I=\displaystyle \int \frac{x^{- 10}}{1 + x^{- 9}}dx$
Put $1+x^{- 9}=t$
$\Rightarrow -9x^{- 10}dx=dt$
$\Rightarrow Ι=\displaystyle \int \frac{d t}{\left(- 9\right) t}$
$=\frac{- 1}{9}ln t+C$
$=\frac{- 1}{9}ln\left(1 + x^{- 9}\right)+C$
$=\frac{1}{9}ln \left(\frac{x^{9}}{1 + x^{9}}\right)+C$
$\Rightarrow \lambda =\frac{1}{9},\mu =9$
$\therefore \frac{1}{\lambda }+\mu =9+9=18$