Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [t / (t⁴ − t² − 2)] dt

Verified step by step guidance

1. Start by examining the integral: \(\int \frac{t}{t^{4} - t^{2} - 2} \, dt\). Notice that the denominator is a quartic polynomial, which might be factorable to simplify the expression.
2. Try to factor the denominator \(t^{4} - t^{2} - 2\). Look for it as a quadratic in terms of \(t^{2}\): let \(u = t^{2}\), then rewrite as \(u^{2} - u - 2\).
3. Factor the quadratic in \(u\): \(u^{2} - u - 2 = (u - 2)(u + 1)\). Substitute back \(u = t^{2}\) to get \(t^{4} - t^{2} - 2 = (t^{2} - 2)(t^{2} + 1)\).
4. Rewrite the integral using the factorization: \(\int \frac{t}{(t^{2} - 2)(t^{2} + 1)} \, dt\). Now consider a substitution to simplify the integral. Since the numerator is \(t\) and the denominator involves \(t^{2}\) terms, try the substitution \(w = t^{2}\), which implies \(dw = 2t \, dt\) or \(t \, dt = \frac{dw}{2}\).
5. Rewrite the integral in terms of \(w\): \(\int \frac{t}{(t^{2} - 2)(t^{2} + 1)} \, dt = \int \frac{1}{(w - 2)(w + 1)} \cdot t \, dt = \int \frac{1}{(w - 2)(w + 1)} \cdot \frac{dw}{2} = \frac{1}{2} \int \frac{1}{(w - 2)(w + 1)} \, dw\). Now the integral is a rational function in \(w\) that can be solved using partial fraction decomposition.

Key Concepts

Integration by Substitution

Integration by substitution is a technique used to simplify integrals by changing variables. It involves identifying a part of the integrand whose derivative also appears in the integral, allowing the integral to be rewritten in terms of a new variable. This method is especially useful when the integral contains composite functions.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of simpler polynomials. This is crucial for simplifying rational expressions in integrals, as it can reveal terms that cancel or suggest substitutions. Recognizing factorable forms helps in breaking down complex denominators or numerators.

Partial Fraction Decomposition

Partial fraction decomposition is a method to express a rational function as a sum of simpler fractions. This technique is often used after factoring the denominator to rewrite the integral into a sum of easier integrals. It is essential for integrating rational functions that cannot be integrated directly.