103. Integración por partes, logaritmo natural (Ejemplo resuelto)

Introduction

This tutorial guides you through the process of integrating the natural logarithm using the method of integration by parts. This technique is essential in calculus, particularly when dealing with logarithmic functions and can significantly simplify complex integrals.

Step 1: Understand the Integration by Parts Formula

Integration by parts is based on the formula:

[ \int u , dv = uv - \int v , du ]

Where:

• ( u ) is a function we choose to differentiate.
• ( dv ) is a function we choose to integrate.
• ( du ) is the derivative of ( u ).
• ( v ) is the integral of ( dv ).

Practical Tips

• Choose ( u ) as the logarithmic part to simplify the integral.
• Choose ( dv ) as the remaining part of the function.

Step 2: Identify Your Functions

For the integral involving the natural logarithm, let's use the following example:

[ \int \ln(x) , dx ]

Selection of Functions

• Set ( u = \ln(x) )
• Set ( dv = dx )

Finding ( du ) and ( v )

• Differentiate ( u ): [ du = \frac{1}{x} , dx ]
• Integrate ( dv ): [ v = x ]

Step 3: Apply the Integration by Parts Formula

Using the identified functions and their derivatives, apply the integration by parts formula:

[ \int \ln(x) , dx = x \ln(x) - \int x \cdot \frac{1}{x} , dx ]

Simplifying the Integral

This simplifies to:

[ \int \ln(x) , dx = x \ln(x) - \int 1 , dx ]

Step 4: Solve the Remaining Integral

Now, solve the remaining integral:

[ \int 1 , dx = x ]

Putting it all together:

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

Where ( C ) is the constant of integration.

Conclusion

You have successfully integrated the natural logarithm using integration by parts. The final result is:

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

Next Steps

• Practice integrating other functions using the same method.
• Explore additional techniques in calculus such as definite integrals and advanced integration methods.

This structured approach not only helps in solving integrals involving logarithms but also strengthens your overall understanding of integration by parts.