In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (3t² + t + 4) / (t³ + t) dt from 1 to √3

Verified step by step guidance

1. First, factor the denominator of the integrand. The denominator is \(t^{3} + t\), which can be factored as \(t(t^{2} + 1)\).
2. Next, express the integrand \(\frac{3t^{2} + t + 4}{t(t^{2} + 1)}\) as a sum of partial fractions. Since the denominator factors into a linear term \(t\) and an irreducible quadratic \(t^{2} + 1\), set up the decomposition as: \(\frac{3t^{2} + t + 4}{t(t^{2} + 1)} = \frac{A}{t} + \frac{Bt + C}{t^{2} + 1}\), where \(A\), \(B\), and \(C\) are constants to be determined.
3. Multiply both sides of the equation by the denominator \(t(t^{2} + 1)\) to clear the fractions, resulting in: \(3t^{2} + t + 4 = A(t^{2} + 1) + (Bt + C)t\).
4. Expand the right-hand side and group like terms: \(3t^{2} + t + 4 = A t^{2} + A + B t^{2} + C t = (A + B) t^{2} + C t + A\). Then, equate the coefficients of corresponding powers of \(t\) on both sides to form a system of equations: for \(t^{2}\), \(3 = A + B\); for \(t\), \(1 = C\); for the constant term, \(4 = A\).
5. Solve the system of equations to find the values of \(A\), \(B\), and \(C\). Once these constants are found, rewrite the integrand as the sum of partial fractions and then integrate each term separately over the interval from \(1\) to \(\sqrt{3}\).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions that are easier to integrate. It involves expressing the integrand as a sum of fractions with simpler denominators, typically linear or quadratic factors. This method is essential when integrating rational functions where the degree of the numerator is less than the degree of the denominator.

Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into a form that matches standard integral formulas. After partial fraction decomposition, each simpler fraction can be integrated using basic rules, such as integrating 1/(t - a) or 1/(t² + b²). Understanding these standard integrals helps in evaluating the definite integral accurately.

Definite Integrals and Limits of Integration

A definite integral calculates the net area under a curve between two points, given by the limits of integration. After finding the antiderivative, the Fundamental Theorem of Calculus is applied by evaluating the antiderivative at the upper and lower limits and subtracting. Correctly handling these limits is crucial for obtaining the final numerical value of the integral.