Question

$\int \frac{dx}{x + x\log x} =$

• A. $\log(1+\log x)+C$
• B. $\log(\log(1+\log x)) + C$
• C. $\log x+\log(\log x)+C$
• D. $\log(1-\log x)+C$

Answer 1

Answer: (A)

$\log(1+\log x)+C$

Answer 2

The integral $\int \frac{dx}{x + x\log x}$ is solved by factoring the denominator as $x(1 + \log x)$. Then, a substitution $u = 1 + \log x$ is made, which implies $du = \frac{1}{x} dx$. This transforms the integral into $\int \frac{du}{u}$, a standard integral whose solution is $\log|u| + C$. Substituting back $u = 1 + \log x$ yields the final result $\log|1 + \log x| + C$.