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

∫₀ᵉ² (ln p)/p dp

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Step 1: Recognize that the integral ∫₀ᵉ² (ln p)/p dp can be solved using a substitution method. Let u = ln(p), which simplifies the logarithmic term.

Step 2: Compute the derivative of u with respect to p. Since u = ln(p), we have du/dp = 1/p, or equivalently, du = (1/p) dp.

Step 3: Substitute u and du into the integral. The integral becomes ∫₀ᵉ² (ln p)/p dp = ∫₀ᵉ² u du.

Step 4: Adjust the limits of integration to match the substitution. When p = 0, ln(p) is undefined, but the integral is defined for positive values approaching 0. When p = e², ln(p) = 2. Thus, the new limits are from u = 0 to u = 2.

Step 5: Evaluate the integral ∫₀² u du using the formula for the integral of u, which is (u²/2). Apply the limits of integration to find the result.

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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 number that quantifies the accumulation of quantities, such as area, over the interval [a, b].

Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integrand by transforming it into a more manageable form. This involves substituting a new variable for an existing one, which can make the integral easier to evaluate. The Jacobian determinant is often used to adjust for the change in variable when dealing with multiple integrals.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base 'e', where 'e' is approximately equal to 2.71828. It is a fundamental function in calculus, particularly in integration and differentiation. The natural logarithm has unique properties, such as ln(ab) = ln(a) + ln(b), which can be useful when simplifying integrals involving logarithmic expressions.

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