7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
24. ∫ dt / √(1 + 4eᵗ)

Verified step by step guidance

1. Step 1: Recognize that the integral ∫ dt / √(1 + 4eᵗ) does not directly match a standard form in the table of integrals. Begin by identifying a substitution that simplifies the expression. Let u = 4eᵗ, which implies du = 4eᵗ dt or dt = du / (4eᵗ).
2. Step 2: Substitute u = 4eᵗ into the integral. The integral becomes ∫ (1 / √(1 + u)) * (du / 4eᵗ). Since eᵗ = u / 4, replace eᵗ with u / 4 in the denominator, simplifying the integral to ∫ (1 / √(1 + u)) * (du / u).
3. Step 3: Simplify the integral further. The expression now becomes (1/4) ∫ (1 / (u√(1 + u))) du. This matches a standard form in the table of integrals, which involves the substitution u = x² or another transformation to simplify the square root.
4. Step 4: Refer to the table of integrals for the standard form that matches ∫ (1 / (u√(1 + u))) du. Use the appropriate formula from the table to rewrite the integral in terms of u.
5. Step 5: After applying the formula from the table, back-substitute u = 4eᵗ to express the result in terms of t. Ensure the final expression is simplified and represents the indefinite integral.

Key Concepts

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration. Understanding how to evaluate indefinite integrals is crucial for solving problems in calculus, as they provide the antiderivative of a function.

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This method is often necessary for simplifying integrals involving quadratic terms, making them easier to integrate. It allows for the identification of standard integral forms in tables.

Integral Tables

Integral tables are collections of standard integrals that provide quick references for evaluating common integrals. They simplify the integration process by allowing students to look up known forms rather than calculating integrals from scratch. Familiarity with these tables is essential for efficiently solving complex integrals.