What is the integral of ln x?
Understand the Problem
The question is asking for the integral of the natural logarithm function, ln(x). To solve this, we will use integration techniques, particularly integration by parts.
Answer
$$ \int \ln(x) \, dx = x \ln(x) - x + C $$
Answer for screen readers
The integral of the natural logarithm function is
$$ \int \ln(x) , dx = x \ln(x) - x + C $$
Steps to Solve
- Set up the integration by parts formula
Integration by parts is given by the formula:
$$ \int u , dv = uv - \int v , du $$
For this problem, we can choose ( u = \ln(x) ) and ( dv = dx ).
- Differentiate and integrate the parts
Next, we need to differentiate ( u ) and integrate ( dv ):
- Differentiate ( u ):
$$ du = \frac{1}{x} , dx $$
- Integrate ( dv ):
$$ v = x $$
- Apply the integration by parts formula
Now we can substitute ( u ), ( du ), ( v ), and ( dv ) into the integration by parts formula:
$$ \int \ln(x) , dx = x \ln(x) - \int x \frac{1}{x} , dx $$
This simplifies to:
$$ \int \ln(x) , dx = x \ln(x) - \int 1 , dx $$
- Integrate the remaining expression
The remaining integral is simple:
$$ \int 1 , dx = x $$
So we can update our expression:
$$ \int \ln(x) , dx = x \ln(x) - x $$
- Add the constant of integration
Finally, we always include the constant of integration, ( C ):
$$ \int \ln(x) , dx = x \ln(x) - x + C $$
The integral of the natural logarithm function is
$$ \int \ln(x) , dx = x \ln(x) - x + C $$
More Information
This integral result is important in various applications across calculus, including solving problems in economics, physics, and engineering. The logarithm function appears frequently, and being able to integrate it is a valuable skill.
Tips
- Forgetting to include the constant of integration ( C ) at the end of the solution.
- Misidentifying ( u ) and ( dv ); choosing inappropriate parts can lead to a more complicated integral.
- Overlooking the simplification of the remaining integral after applying integration by parts.
