Integration of Logarithmic Functions: A Comprehensive Guide

When dealing with complex functions, particularly those involving logarithms, integration by parts is a powerful technique. In this article, we will provide a detailed step-by-step guide to integrating the function log x / x1^2 using integration by parts and partial fraction decomposition.

1. Introduction to the Problem

We need to evaluate the integral:

( ∫frac{log x}{x^{2}}, dx )

2. Step-by-Step Solution

Step 1: Choose u and dv

To apply the integration by parts formula, we start by setting:

( u log x Rightarrow du frac{1}{x}dx )

( dv frac{1}{x^2}dx Rightarrow v -frac{1}{x} )

Step 2: Apply Integration by Parts

Using the formula:

( ∫u,dv uv - ∫v,du )

We get:

( ∫frac{log x}{x^2}, dx -frac{log x}{x} - ∫ -frac{1}{x} cdot frac{1}{x}, dx )

This simplifies to:

( -frac{log x}{x} ∫ frac{1}{x^2}, dx )

3. Simplify the Remaining Integral

Now, we need to integrate 1 / (x x^2). We can use partial fraction decomposition:

( frac{1}{x x^2} frac{A}{x} - frac{B}{x^2} )

Multiplying through by x x^2 gives:

( 1 Ax^2 - Bx )

Setting up the equations:

For x 0: ( 1 A cdot 0 - B cdot 0 ) rarr; A 1

For x 1: ( 1 B cdot 1 ) rarr; B -1

Thus, we have:

( frac{1}{x x^2} frac{1}{x} - frac{1}{x^2} )

4. Integrate

Now we can integrate:

( ∫left( frac{1}{x} - frac{1}{x^2} right) dx log x - log x^2 C )

This simplifies to:

( log left( frac{x}{x^2} right) C log left( frac{1}{x} right) C )

Therefore:

( ∫ frac{log x}{x^2}, dx -frac{log x}{x} log x C )

5. Final Answer

Putting it all together, we get:

( ∫ frac{log x}{x^2}, dx -frac{log x}{x} log x C )

Final Answer:

( ∫ frac{log x}{x^2}, dx -frac{log x}{x} log x C )

Conclusion

In this article, we have demonstrated how to integrate a logarithmic function using integration by parts and partial fraction decomposition. The key steps involve breaking down the integral into manageable parts and solving each step methodically.

Keywords

Integration by parts Logarithmic functions Partial fraction decomposition