Textbook Question
41–48. Geometry problems Use a table of integrals to solve the following problems.
46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.2.11
9–40. Integration by parts Evaluate the following integrals using integration by parts.
11. ∫ t · e⁶ᵗ dt
Verified step by step guidance1
Identify the parts of the integral for integration by parts. Let \(u = t\) and \(dv = e^{6t} dt\).
Compute the derivatives and integrals needed: find \(du = dt\) and integrate \(dv\) to get \(v = \frac{1}{6} e^{6t}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Substitute the expressions for \(u\), \(v\), \(du\) into the formula: \(\int t e^{6t} dt = t \cdot \frac{1}{6} e^{6t} - \int \frac{1}{6} e^{6t} dt\).
Evaluate the remaining integral \(\int e^{6t} dt\) and simplify the expression to write the final integral in terms of \(t\) and \(e^{6t}\) plus the constant of integration.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts Formula
Integration by parts is a technique derived from the product rule of differentiation. It states that ∫u dv = uv - ∫v du, where u and dv are parts of the original integral chosen to simplify the problem. Selecting u and dv wisely is crucial for easier integration.
Recommended video:
08:30
Introduction to Integration by Parts
Choosing u and dv
In integration by parts, u is typically chosen as a function that simplifies when differentiated, while dv is chosen as a function that is easy to integrate. For example, in ∫ t · e^(6t) dt, choosing u = t and dv = e^(6t) dt helps reduce the integral complexity.
Recommended video:
07:51Choosing a Convergence Test
Integrating Exponential Functions
Integrating exponential functions like e^(kt) involves dividing by the constant k after integration, resulting in (1/k) e^(kt). This property is essential when integrating dv in problems involving exponentials, ensuring correct evaluation of the integral.
Recommended video:
05:11Integrals of General Exponential Functions
Related Practice
Textbook Question
7–84. Evaluate the following integrals.
49. ∫ tan³x · sec⁹x dx
Textbook Question
7–64. Integration review Evaluate the following integrals.
49. ∫ √(9 + √(t + 1)) dt
Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
39. ∫ x²/(100 - x²)^(3/2) dx
Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
26. ∫[√2 to √2] √(x² - 1)/x dx
Textbook Question
15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.
15. ∫(2 to 10) 2x² dx using n = 1, 2, and 4 subintervals