91Ó°ÊÓ

Partial fraction decomposition is a useful algebraic method used for breaking down complex rational expressions into simpler terms that can be more easily integrated. In this exercise, the integrand \( \frac{2}{t^2 - 1} \) is given in a form that can be simplified by recognizing the denominator as a difference of squares: \( t^2 - 1 = (t - 1)(t + 1) \). This structure indicates that the integrand can be expressed as the sum of two simpler fractions.To achieve this, we find constants \( A \) and \( B \) such that:

• \( \frac{2}{t^2 - 1} = \frac{A}{t - 1} + \frac{B}{t + 1} \)

By matching coefficients after expanding \( 2 = A(t + 1) + B(t - 1) \), the constants are determined to be \( A = 1 \) and \( B = -1 \). This simplifies the integrand to:

• \( \frac{1}{t - 1} - \frac{1}{t + 1} \)

This decomposition allows for simpler integration in the following steps.

A definite integral calculates the accumulated value from a specific start point to an end point. It has both an upper and lower bound, signifying where the accumulation begins and ends. Here, the given integral is \( \int_{2}^{\infty} \frac{2}{t^2 - 1} \, dt \) which specifies boundaries from 2 to infinity.The integration process involves evaluating the integral:

• \( \int \left( \frac{1}{t - 1} - \frac{1}{t + 1} \right) \, dt \)

Each part is integrated separately:

• \( \int \frac{1}{t - 1} \, dt = \ln|t - 1| + C_1 \)
• \( \int -\frac{1}{t + 1} \, dt = -\ln|t + 1| + C_2 \)

These expressions combine to form the result:

• \( \ln \left| \frac{t - 1}{t + 1} \right| \)

Applying the limits from \( t = 2 \) to \( t = \infty \), results in evaluating the expression at these boundaries to find the definite integral's value.

Convergence is a critical concept when dealing with integrals that have infinite limits. It determines whether the integral results in a finite value or not. For an improper integral like \( \int_{2}^{\infty} \frac{2}{t^2 - 1} \, dt \), checking convergence involves considering the behavior as the upper limit approaches infinity.The expression \( \ln \left| \frac{b - 1}{b + 1} \right| \) as \( b \to \infty \) moves towards zero, since the fraction approaches 1, and \( \ln 1 = 0 \). This indicates that the integral converges.Thus, the definite integral presents a finite value:

• \(- \ln \left( \frac{1}{3} \right) = \ln 3 \)

Convergence confirmation means that the total accumulation from 2 to infinity is not only calculable, but ultimately finite, resulting in a value of \( \ln 3 \).