Evaluating integrals Evaluate the following integrals.

∫₁ᵉ d𝓍 / [𝓍(1 + ln 𝓍)]

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Step 1: Recognize that the integral ∫₁ᵉ d𝓍 / [𝓍(1 + ln 𝓍)] involves a logarithmic function in the denominator, suggesting a substitution method might simplify the problem. Let u = ln(𝓍), which implies that du = d𝓍 / 𝓍.

Step 2: Substitute u = ln(𝓍) into the integral. When 𝓍 = 1, u = ln(1) = 0, and when 𝓍 = e, u = ln(e) = 1. The integral now becomes ∫₀¹ du / (1 + u).

Step 3: Recognize that the integral ∫₀¹ du / (1 + u) is a standard form that can be solved using the natural logarithm function. Specifically, the integral of 1 / (1 + u) is ln|1 + u|.

Step 4: Apply the antiderivative formula to evaluate the integral. The result is ln|1 + u| evaluated from u = 0 to u = 1.

Step 5: Substitute the limits of integration into the antiderivative expression ln|1 + u| to complete the evaluation. Simplify the result to express the final answer.

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Key Concepts

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

Integration

Integration is a fundamental concept in calculus that involves finding the accumulated area under a curve represented by a function. It is the reverse process of differentiation and can be used to calculate quantities such as areas, volumes, and total accumulated change. The integral symbol (∫) denotes the operation, and definite integrals have specified limits, while indefinite integrals do not.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a crucial function in calculus, particularly in integration and differentiation, as it arises in various contexts, including growth processes and compound interest. Understanding its properties, such as ln(ab) = ln(a) + ln(b), is essential for manipulating expressions involving logarithms.

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing the variable of integration. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative.

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