Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.

∫₁³ ( 2ˣ / 2ˣ + 4 ) d𝓍

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Step 1: Observe the integral ∫₁³ (2ˣ / (2ˣ + 4)) d𝓍. To simplify this, consider using a substitution method. Let u = 2ˣ + 4, which simplifies the denominator.

Step 2: Differentiate u with respect to x. Since u = 2ˣ + 4, du/dx = ln(2) * 2ˣ. Rearrange to express dx in terms of du: dx = du / (ln(2) * 2ˣ).

Step 3: Substitute u and dx into the integral. Replace 2ˣ in the numerator and denominator using the substitution u = 2ˣ + 4. The integral becomes ∫ (1 / u) * (du / ln(2)).

Step 4: Factor out constants from the integral. The constant 1/ln(2) can be factored out, leaving (1/ln(2)) ∫ (1/u) du.

Step 5: Evaluate the integral of 1/u with respect to u. The result is ln|u|. Substitute back u = 2ˣ + 4 and apply the limits of integration (x = 1 to x = 3) to find the definite integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫_a^b f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b].

Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integrand. By substituting a new variable for an existing one, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with complex functions or when the limits of integration need to be adjusted accordingly.

Integration Techniques

Integration techniques encompass various methods used to evaluate integrals, including substitution, integration by parts, and using tables of integrals. These techniques help in solving integrals that may not be straightforward. Familiarity with these methods, such as those found in Table 5.6, allows for efficient evaluation of definite integrals in calculus.

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