Textbook Question
Evaluate the integrals in Exercises 33–54.
∫(e^(3x) + 5e^(-x)) dx
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.97
Evaluate the integrals in Exercises 97–110.
97. ∫ 3x^(√3) dx
Verified step by step guidance1
Identify the integral to be solved: \(\int 3x^{\sqrt{3}} \, dx\).
Recall the power rule for integration: for any real number \(n \neq -1\), \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\).
Since the integrand is \(3x^{\sqrt{3}}\), factor out the constant 3: \(3 \int x^{\sqrt{3}} \, dx\).
Apply the power rule to \(\int x^{\sqrt{3}} \, dx\), which gives \(\frac{x^{\sqrt{3} + 1}}{\sqrt{3} + 1} + C\).
Multiply the result by 3 to get the integral: \(3 \times \frac{x^{\sqrt{3} + 1}}{\sqrt{3} + 1} + C\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Power Functions
Integrating power functions involves applying the rule ∫ x^n dx = (x^(n+1)) / (n+1) + C, where n ≠ -1. This formula is fundamental for integrating expressions where the variable is raised to a constant exponent.
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Representing Functions as Power Series
Handling Irrational Exponents
When the exponent is an irrational number, such as √3, the integration process remains the same as with rational exponents. Treat the exponent as a constant and apply the power rule accordingly.
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Guided course
7:39Introduction to Exponent Rules
Constant Multipliers in Integration
Constants multiplied by functions can be factored out of the integral. For example, ∫ 3x^n dx = 3 ∫ x^n dx, simplifying the integration process by focusing on the variable part.
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7:24Multiplying & Dividing Functions
Related Practice
Textbook Question
Evaluate the integrals in Exercises 33–54.
∫ (e^(1/x) / x²) dx
Textbook Question
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
33. y = csch⁻¹(1/2)^θ
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
81. y = log₁₀(e^x)
Textbook Question
In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
3. lim (x → ∞) (5x² - 3x) / (7x² + 1)
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Textbook Question
Evaluate the integrals in Exercises 41–60.
43. ∫6cosh(x/2 - ln3)dx