What is the integration of ln x?
Understand the Problem
The question is asking for the integration of the natural logarithm function, ln(x). This requires knowledge of calculus, specifically integration techniques.
Answer
$$ \int \ln(x) \, dx = x \ln(x) - x + C $$
Answer for screen readers
The integral of the natural logarithm function is given by:
$$ \int \ln(x) , dx = x \ln(x) - x + C $$
Steps to Solve
- Set up the integral
We need to find the integral of the natural logarithm function, which can be written as:
$$ \int \ln(x) , dx $$
- Use integration by parts
To solve this integral, we can use the integration by parts formula:
$$ \int u , dv = uv - \int v , du $$
We will choose:
- ( u = \ln(x) ), which means ( du = \frac{1}{x} , dx )
- ( dv = dx ), which means ( v = x )
- Apply the integration by parts formula
Now, substituting our choices into the integration by parts formula:
$$ \int \ln(x) , dx = x \ln(x) - \int x \cdot \frac{1}{x} , dx $$
This simplifies to:
$$ \int \ln(x) , dx = x \ln(x) - \int 1 , dx $$
- Integrate the remaining part
Now we can integrate the remaining part:
$$ \int 1 , dx = x $$
- Combine the results
Putting it all back together:
$$ \int \ln(x) , dx = x \ln(x) - x + C $$
Where ( C ) is the constant of integration.
The integral of the natural logarithm function is given by:
$$ \int \ln(x) , dx = x \ln(x) - x + C $$
More Information
This result is important in calculus and is frequently used in various applications, including areas such as economics and physics, where logarithmic functions arise often. It’s a common integral that shows the relationship between logarithmic and polynomial functions.
Tips
- Forgetting to include the constant of integration ( C ) after performing the indefinite integral.
- Confusing the variable of integration; make sure to differentiate ( x ) properly as it represents the integration variable.
- Not applying the integration by parts formula correctly by misidentifying ( u ) and ( dv ).
