Question

If the integral $I=\int \frac{dx}{x^{10}+x}=$ $\lambda ln\left(\frac{x^9}{1+x^{\mu }}\right)+C,$ (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)

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