Evaluate the integrals in Exercises 31–78.
77. ∫dt/((t+1)√(t²+2t-8))

Verified step by step guidance

1. Start by examining the integral: \(\int \frac{dt}{(t+1) \sqrt{t^{2} + 2t - 8}}\). Notice the expression under the square root, \(t^{2} + 2t - 8\), which can be simplified by completing the square.
2. Complete the square for the quadratic inside the square root: \(t^{2} + 2t - 8 = (t^{2} + 2t + 1) - 1 - 8 = (t + 1)^{2} - 9\). Rewrite the integral as \(\int \frac{dt}{(t+1) \sqrt{(t+1)^{2} - 9}}\).
3. Make the substitution \(x = t + 1\), so that \(dt = dx\). The integral becomes \(\int \frac{dx}{x \sqrt{x^{2} - 9}}\). This substitution simplifies the integral and prepares it for a trigonometric substitution.
4. Use a trigonometric substitution to handle the square root: since \(\sqrt{x^{2} - 9}\) resembles \(\sqrt{x^{2} - a^{2}}\), set \(x = 3 \sec \theta\), which implies \(dx = 3 \sec \theta \tan \theta \, d\theta\) and \(\sqrt{x^{2} - 9} = 3 \tan \theta\).
5. Rewrite the integral in terms of \(\theta\): substitute \(x\), \(dx\), and \(\sqrt{x^{2} - 9}\) into the integral to get \(\int \frac{3 \sec \theta \tan \theta \, d\theta}{3 \sec \theta \cdot 3 \tan \theta}\). Simplify the expression and then integrate with respect to \(\theta\). After integration, back-substitute to express the answer in terms of \(t\).

Key Concepts

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By identifying a part of the integrand as a new variable, the integral can be transformed into a more manageable form, often reducing complex expressions involving roots or polynomials.

Completing the Square

Completing the square rewrites quadratic expressions into a perfect square plus or minus a constant. This technique is useful for simplifying expressions under square roots, making it easier to apply substitution or recognize standard integral forms.

Standard Integrals Involving Square Roots

Certain integrals involving square roots of quadratic expressions have known standard forms, such as those involving arcsine, logarithmic, or inverse hyperbolic functions. Recognizing these forms helps in evaluating integrals efficiently after appropriate algebraic manipulation.