Question

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

Accepted Answer

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. 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. 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 $$ t . $$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. 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 $$ \int \frac{1}{t} dt .$$

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

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Concetti chiave

Integration of Rational Functions

Partial Fraction Decomposition

Definite Integrals and Limits of Integration