Integrate 1/x ln x
Understand the Problem
The question is asking us to find the integral of the function 1/(x ln(x)). This implies we need to apply techniques of integration to solve this problem.
Answer
$$ \ln|\ln(x)| + C $$
Answer for screen readers
The final answer is
$$ \ln|\ln(x)| + C $$
Steps to Solve
- Identify the Integral to Solve
We need to evaluate the integral
$$ \int \frac{1}{x \ln(x)} , dx $$
- Use Substitution
Let's use the substitution method. Set
$$ u = \ln(x) $$
Then we differentiate to find $dx$ in terms of $du$:
$$ du = \frac{1}{x} , dx \implies dx = x , du $$
Since $x = e^u$ from our substitution, we get
$$ dx = e^u , du $$
- Change the Variables in the Integral
Now substitute $u$ and $dx$ into the integral:
$$ \int \frac{1}{x \ln(x)} , dx = \int \frac{1}{e^u \cdot u} \cdot e^u , du $$
This simplifies to:
$$ \int \frac{1}{u} , du $$
- Evaluate the Integral
The integral
$$ \int \frac{1}{u} , du $$
is a standard integral that yields:
$$ \ln|u| + C $$
where $C$ is the constant of integration.
- Substitute Back for Original Variable
Now, convert back to the variable $x$. Recall that $u = \ln(x)$, so we replace $u$:
$$ \ln|\ln(x)| + C $$
The final answer is
$$ \ln|\ln(x)| + C $$
More Information
This integral is a common technique in calculus, especially in problems involving logarithmic functions. The substitution method simplifies the process by rephrasing the integral in terms of $u$, which is easier to handle.
Tips
- Forgetting to substitute back to the original variable after integration.
- Not accounting for absolute values in logarithmic functions when finding $u$.
