Question

$\int\frac{1}{x(\log x)}dx=$

• A. $\log|\log x|+C$
• B. $\frac{(\log|x|)^{2}}{2}+C$
• C. $\log|x|+C$
• D. $\frac{1}{\log|x|}+C$
• E. $\frac{1}{(\log|x|)^{2}}+C$

Hint:

Calculus Tip: The pattern $\int \frac{f^{\prime}(x)}{f(x)} dx = \log|f(x)| + C$ is universally true. Here, $f(x) = \log x$, making the evaluation instantaneous without writing out the full substitution.

Answer 1

The Correct Option is A

Concept:
This integral is solved using the method of substitution (u-substitution). When an integrand contains a function and its exact derivative, substituting a new variable for that function simplifies the integral into a basic form.

Step 1: Identify the function and its derivative.
The integrand is $\frac{1}{x \log x}$, which can be rewritten to separate the terms: $$I = \int \frac{1}{\log x} \cdot \frac{1}{x} dx$$ Notice that the derivative of $\log x$ is exactly $\frac{1}{x}$.

Step 2: Perform the u-substitution.
Let $u = \log x$. Differentiate both sides with respect to $x$: $$\frac{du}{dx} = \frac{1}{x} \implies du = \frac{1}{x} dx$$

Step 3: Rewrite the integral in terms of u.
Substitute $u$ and $du$ back into the separated integral from
Step 1: $$I = \int \frac{1}{u} du$$

Step 4: Integrate the simplified expression.
The integral of $\frac{1}{u}$ is a standard elementary form: $$I = \log|u| + C$$

Step 5: Substitute x back into the equation.
Replace the intermediate variable $u$ with its original definition ($\log x$): $$I = \log|\log x| + C$$ Hence the correct answer is (A) $\log|\log x|+C$.