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.
40. ∫ (e³ᵗ / √(4 + e²ᵗ)) dt

Guida verificata passo dopo passo

Step 1: Recognize that the integral ∫ (e³ᵗ / √(4 + e²ᵗ)) dt involves a combination of exponential and square root terms. To simplify, consider a substitution to reduce the complexity of the integrand.

Step 2: Let u = e²ᵗ. Then, compute the derivative of u with respect to t: du/dt = 2e²ᵗ. Rearrange to express dt in terms of du: dt = du / (2e²ᵗ).

Step 3: Substitute u = e²ᵗ into the integral. The term e³ᵗ can be rewritten as e²ᵗ * eᵗ, and e²ᵗ is replaced by u. The integral becomes ∫ (u * eᵗ / √(4 + u)) * (du / (2u)).

Step 4: Simplify the expression. Notice that eᵗ = √u (since u = e²ᵗ). Replace eᵗ with √u, and cancel out terms where possible. The integral now becomes ∫ (√u / √(4 + u)) * (du / 2).

Step 5: Factor out constants and rewrite the integral in a form that matches a standard table of integrals. The integral becomes (1/2) ∫ (√u / √(4 + u)) du. Use the table of integrals to find the solution or proceed with further simplifications if necessary.

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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, typically denoted as 'C'. Understanding how to evaluate indefinite integrals is crucial for solving problems in calculus, as they provide the antiderivative of a function.

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