Textbook Question
Multiple substitutions If necessary, use two or more substitutions to find the following integrals.
∫ 𝓍 sin⁴ 𝓍² cos 𝓍² d𝓍 (Hint: Begin with u = 𝓍², and then use v = sin u .)
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
1:47 minutesProblem 8.1.65d
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. ∫(1/eˣ) dx = ln eˣ + C.
Verified step by step guidance1
Step 1: Analyze the given integral ∫(1/eˣ) dx. The integrand is 1/eˣ, which can be rewritten as e⁻ˣ using the property of exponents.
Step 2: Recall the standard rule for integrating exponential functions. The integral of e⁻ˣ with respect to x is ∫e⁻ˣ dx = -e⁻ˣ + C, where C is the constant of integration.
Step 3: Compare the given statement ∫(1/eˣ) dx = ln eˣ + C with the correct integral result. Note that ln eˣ simplifies to x because the natural logarithm and exponential functions are inverses of each other.
Step 4: Observe that the given statement suggests the integral of 1/eˣ is ln eˣ + C, which simplifies to x + C. This is incorrect because the integral of 1/eˣ is actually -e⁻ˣ + C, not x + C.
Step 5: Conclude that the statement ∫(1/eˣ) dx = ln eˣ + C is false. Provide the correct result, ∫(1/eˣ) dx = -e⁻ˣ + C, as a counterexample to demonstrate why the given statement is incorrect.
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1mKey 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 antiderivative of a function. It is the process of determining the area under a curve represented by a function over a specified interval. Understanding integration is crucial for evaluating definite and indefinite integrals, which are essential for solving problems related to areas, volumes, and other applications in mathematics and physics.
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Natural Logarithm
The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is an important function in calculus, particularly in integration and differentiation. The natural logarithm has unique properties, such as ln(e^x) = x, which are useful for simplifying expressions and solving equations involving exponential functions.
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Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where a and b are constants, and e is the base of the natural logarithm. These functions are characterized by their rapid growth or decay and are widely used in various fields, including finance, biology, and physics. Understanding the properties of exponential functions is essential for evaluating integrals involving e^x and applying the rules of integration correctly.
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