Use Table 5.6 to evaluate the following indefinite integrals.

(a) ∫ e¹⁰ˣ d𝓍

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Step 1: Recognize that the integral involves the exponential function e raised to a linear term (10x). The formula for the integral of e^(kx) is (1/k) * e^(kx) + C, where k is a constant and C is the constant of integration.

Step 2: Identify the constant k in the exponent. In this case, k = 10 because the exponent is 10x.

Step 3: Apply the formula for the integral of e^(kx). Substitute k = 10 into the formula, resulting in (1/10) * e^(10x) + C.

Step 4: Write the result in terms of the indefinite integral. The integral ∫ e^(10x) dx simplifies to (1/10) * e^(10x) + C.

Step 5: Remember to include the constant of integration (C) in your final answer, as this is an indefinite integral.

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Key Concepts

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

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed with the integral sign followed by the function and the differential, and they include a constant of integration (C) since the derivative of a constant is zero. Understanding how to evaluate indefinite integrals is crucial for solving problems in calculus, as they provide the antiderivative of a function.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. In calculus, the natural exponential function e^x is particularly important due to its unique property that the derivative of e^x is e^x itself. This property simplifies the process of integration, especially when dealing with integrals involving exponential terms.

Integration Techniques

Integration techniques are methods used to evaluate integrals that may not be straightforward. Common techniques include substitution, integration by parts, and using integral tables. Familiarity with these techniques allows students to tackle a variety of integrals, including those involving exponential functions, and is essential for effectively solving calculus problems.

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