Domanda del libro di testo
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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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.2.2
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ θ cos(πθ) dθ
Guida verificata passo dopo passo1
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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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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Percorso guidato
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.
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Percorso guidato
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.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Pratica correlata
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Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
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Domanda del libro di testo
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Domanda del libro di testo
Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.
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Domanda del libro di testo
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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Domanda del libro di testo
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