Understanding and Evaluating Improper Integrals: Convergence and Divergence

Improper integrals involve integrands that are not defined at a point within the interval of integration or where the integral extends to infinity. In this article, we will evaluate the convergence and divergence of a specific improper integral and address common issues encountered when solving such integrals.

Introduction

The integral in question is:

(int_0^2 frac{dx}{2x - x^2})

First, let's simplify the expression in the denominator:

(2x - x^2 x(2 - x))

This allows us to rewrite the integral as:

(int_0^2 frac{dx}{x(2 - x)})

Evaluation of the Improper Integral

The integral is improper due to the undefined behavior of the integrand at the points (x 0) and (x 2). Therefore, we need to evaluate the integral as a limit:

(int_0^2 frac{dx}{x(2 - x)} lim_{epsilon to 0^ } left(int_epsilon^{2 - delta} frac{dx}{x(2 - x)}right))

where (delta to 0^ ) as well.

Breaking the Integral into Two Parts

To simplify the integration, we can break the integral into two parts:

(int_0^2 frac{dx}{x(2 - x)} lim_{epsilon to 0^ } left(int_epsilon^1 frac{dx}{x(2 - x)} int_1^{2 - delta} frac{dx}{x(2 - x)}right))

Partial Fraction Decomposition

We can decompose the integrand using partial fractions:

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

Multiplying through by the denominator (x(2 - x)) gives:

(1 A(2 - x) Bx)

Setting (x 0) gives:

(A frac{1}{2})

Setting (x 2) gives:

(B frac{1}{2})

So:

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

Evaluating Each Part

Part 1:

(int_epsilon^1 frac{dx}{x(2 - x)} frac{1}{2} int_epsilon^1 left(frac{1}{x} frac{1}{2 - x}right) dx)

Integrating term-by-term:

(frac{1}{2} left[ln|x| - ln|2 - x|right]_epsilon^1)

Calculating the limits:

(x 1):

(frac{1}{2} (ln|1| - ln|1|) 0)

(x epsilon):

(frac{1}{2} (ln|epsilon| - ln|2 - epsilon|) frac{1}{2} lnleft|frac{epsilon}{2 - epsilon}right|)

As (epsilon to 0^ ), (ln|epsilon| to -infty), so this part diverges.

Part 2:

(int_1^{2 - delta} frac{dx}{x(2 - x)} frac{1}{2} int_1^{2 - delta} left(frac{1}{x} frac{1}{2 - x}right) dx)

Integrating term-by-term:

(frac{1}{2} left[ln|x| - ln|2 - x|right]_1^{2 - delta})

Calculating the limits:

(x 2 - delta):

(frac{1}{2} (ln|2 - delta| - ln|delta|) to infty) as (delta to 0^ )

Conclusion

Since both parts of the integral diverge, we conclude that the integral (int_0^2 frac{dx}{2x - x^2}) diverges. This highlights the importance of careful analysis in determining the convergence or divergence of improper integrals.

Key Takeaways:

Improper integrals require integration over a finite or infinite interval where the integrand is undefined. The improper integral is evaluated by taking limits as the bounds approach the singular points. Partial fraction decomposition can be used to simplify complex integrals.

Understanding these concepts is crucial for mastering the evaluation of improper integrals and is essential for advanced calculus and real analysis.