Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (x² / (x² + 1)) dx
Ch. 8 - Techniques of Integration
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 8, Problem 8.2.2
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ θ cos(πθ) dθ
Verified step by step guidance1
Identify the parts of the integral for integration by parts. Let \(u = \theta\) and \(dv = \cos(\pi \theta) \, d\theta\).
Compute the derivatives and antiderivatives needed: \(du = d\theta\) and \(v = \int \cos(\pi \theta) \, d\theta\). To find \(v\), recall that \(\int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + C\), so here \(v = \frac{1}{\pi} \sin(\pi \theta)\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Substitute the expressions found: \(\int \theta \cos(\pi \theta) \, d\theta = \theta \cdot \frac{1}{\pi} \sin(\pi \theta) - \int \frac{1}{\pi} \sin(\pi \theta) \, d\theta\).
Simplify the integral: \(\int \frac{1}{\pi} \sin(\pi \theta) \, d\theta = \frac{1}{\pi} \int \sin(\pi \theta) \, d\theta\). Use the integral formula \(\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C\) to evaluate this integral.
Combine all parts to write the expression for the original integral, including the constant of integration \(C\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts
Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely simplifies the problem.
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06:18
Integration by Parts for Definite Integrals
Choosing u and dv
Selecting which part of the integrand to assign as u and which as dv is crucial. Typically, u is chosen as a function that simplifies when differentiated, and dv as a function that is easy to integrate. This choice affects the ease of solving the integral.
Recommended video:
07:51Choosing a Convergence Test
Integrating Trigonometric Functions
Integrating functions like cos(πθ) involves recognizing standard integral forms. The integral of cos(kθ) with respect to θ is (1/k)sin(kθ) + C, where k is a constant. This knowledge helps in computing the integral of dv in integration by parts.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(1 - (ln x)²) / (x ln x) dx
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Textbook Question
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
85. Find the volume of the solid generated by revolving the region about the y-axis.
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Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 4 to ∞ of (dx / (√x - 1))
Textbook Question
Use any method to evaluate the integrals in Exercises 65–70.
∫ cot(x) / cos²(x) dx
Textbook Question
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
83. Find the area of the region.
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