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

$$ \int \ln(x) \, dx = x \ln(x) - x + C $$

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 ).

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