Solving and Decomposing Indefinite Integrals
This article aims to clarify how to solve integrals that can appear ambiguous, using the specific example given. The importance of clear notation and the technique of partial fractions in integration will be demonstrated through detailed step-by-step processes.
Understanding the Problem
The given problem is:
[int frac{1}{x-1x-2} dx]This expression is ambiguous due to the multiplication in the denominator. We need to determine whether it should be interpreted as:
[int frac{1}{(x-1)(x-2)} dx]or
[int frac{1}{x-1}x-2 dx]Each of these scenarios requires a different approach. Let's solve both cases step by step.
Case 1: (frac{1}{(x-1)(x-2)})
We will use the method of partial fractions. The expression can be decomposed as:
[frac{1}{(x-1)(x-2)} frac{A}{x-1} frac{B}{x-2}]Solving for (A) and (B), we get:
[frac{1}{(x-1)(x-2)} frac{1}{(x-1)(x-2)} frac{1}{x-2} - frac{1}{x-1}]Therefore, we can write the integral as:
[int frac{1}{(x-1)(x-2)} dx int left(frac{1}{x-2} - frac{1}{x-1}right) dx]Integrating each term separately:
[int left(frac{1}{x-2} - frac{1}{x-1}right) dx ln|x-2| - ln|x-1| C]Using the logarithm properties, this simplifies to:
[lnleft|frac{x-2}{x-1}right| C]Case 2: (frac{1}{x-1}x-2)
Here, we need to interpret the expression (frac{1}{x-1}x-2) as (frac{x-2}{x-1}).
We can rewrite this as:
[frac{1}{x-1}x-2 frac{x-1-1}{x-1} 1 - frac{1}{x-1}]Therefore, the integral becomes:
[int left(1 - frac{1}{x-1}right) dx]Integrating term by term:
[int left(1 - frac{1}{x-1}right) dx x - ln|x-1| C]Conclusion
To solve integrals, it is crucial to ensure that the expression is clearly defined. In the first case, the solution is:
[int frac{1}{(x-1)(x-2)} dx lnleft|frac{x-2}{x-1}right| C]While in the second case, the solution is:
[int frac{x-2}{x-1} dx x - ln|x-1| C]Summary
The techniques illustrated here—such as partial fractions and simplification—offer robust methods for solving complex integrals. Proper notation and clear understanding of the problem statement are essential for accurate solutions.
Keywords
- Indefinite integral
- Partial fractions
- Integration