23-64. Integration Evaluate the following integrals.
44. ∫₁² 2/[t³(t + 1)] dt

Verified step by step guidance

1. Start by examining the integral \( \int_{1}^{2} \frac{2}{t^{3}(t+1)} \, dt \). Notice that the integrand is a rational function, which suggests using partial fraction decomposition to simplify it.
2. Set up the partial fraction decomposition for \( \frac{2}{t^{3}(t+1)} \) as follows: \[ \frac{2}{t^{3}(t+1)} = \frac{A}{t} + \frac{B}{t^{2}} + \frac{C}{t^{3}} + \frac{D}{t+1} \]. The goal is to find constants \( A, B, C, D \) that satisfy this identity.
3. Multiply both sides of the equation by the common denominator \( t^{3}(t+1) \) to clear the fractions: \[ 2 = A t^{2}(t+1) + B t (t+1) + C (t+1) + D t^{3} \]. Then expand and collect like terms in powers of \( t \).
4. Equate the coefficients of corresponding powers of \( t \) on both sides to form a system of equations for \( A, B, C, D \). Solve this system to find the values of these constants.
5. Rewrite the integral as the sum of simpler integrals using the found constants: \[ \int_{1}^{2} \left( \frac{A}{t} + \frac{B}{t^{2}} + \frac{C}{t^{3}} + \frac{D}{t+1} \right) dt \]. Then integrate each term separately using standard integral formulas, such as \( \int t^{n} dt \) and \( \int \frac{1}{t} dt \).

Key Concepts

Integration of Rational Functions

This involves integrating functions expressed as ratios of polynomials. Such integrals often require algebraic manipulation, like partial fraction decomposition, to rewrite the integrand into simpler terms that are easier to integrate.

Partial Fraction Decomposition

A technique used to break down complex rational expressions into a sum of simpler fractions. This method simplifies the integration process by allowing each term to be integrated individually using basic integral formulas.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two points. Understanding how to apply the limits of integration after finding the antiderivative is essential to evaluate the integral's exact value.