Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
51. ∫ x²/√(4 + x²) dx
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 8.2.26
9–40. Integration by parts Evaluate the following integrals using integration by parts.
26. ∫ t³ sin(t) dt
Verified step by step guidance1
Identify the integral to solve: \(\int t^{3} \sin(t) \, dt\).
Choose parts for integration by parts: let \(u = t^{3}\) (which simplifies when differentiated) and \(dv = \sin(t) \, dt\) (which is easy to integrate).
Compute the derivatives and integrals needed: find \(du = 3t^{2} \, dt\) and \(v = -\cos(t)\) (since the integral of \(\sin(t)\) is \(-\cos(t)\)).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), so write \(\int t^{3} \sin(t) \, dt = -t^{3} \cos(t) - \int (-\cos(t)) (3t^{2}) \, dt\).
Simplify the integral and prepare to repeat integration by parts on \(\int 3t^{2} \cos(t) \, dt\) to fully evaluate the original integral.
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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 integrand chosen to simplify the integral. Selecting u and dv wisely is crucial for solving complex integrals.
Recommended video:
08:30
Introduction to Integration by Parts
Choosing u and dv
When applying integration by parts, u is typically chosen as a function that simplifies upon differentiation, while dv is chosen as a function that is easy to integrate. For example, in ∫t³ sin(t) dt, choosing u = t³ and dv = sin(t) dt helps reduce the power of t through differentiation.
Recommended video:
07:51Choosing a Convergence Test
Repeated Application of Integration by Parts
Some integrals, like ∫t³ sin(t) dt, require applying integration by parts multiple times to fully evaluate. Each application reduces the power of t until the integral becomes straightforward. Keeping track of each step carefully ensures correct and complete solutions.
Recommended video:
08:23Repeated Integration by Parts
Related Practice
Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
33. ∫ √(x² - 9)/x dx, x > 3
Textbook Question
Evaluate the following integrals.
∫ e³ˣ/(eˣ - 1) dx
Textbook Question
41–48. Geometry problems Use a table of integrals to solve the following problems.
42. Find the length of the curve y = x^(3/2) + 8 on the interval from 0 to 2.
Textbook Question
7–84. Evaluate the following integrals.
47. ∫ [(2x³ + x² - 2x - 4) / (x² - x - 2)] dx
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Textbook Question
29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
33. ∫(0 to π) sin x cos(3x) dx = 0
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