Domanda del libro di testo
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (4 - x²)^(3/2) from 0 to 1
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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 8.2.70
In Exercises 67–73, use integration by parts to establish the reduction formula.
∫ (ln x)^n dx = x (ln x)^n - n ∫ (ln x)^(n-1) dx
Guida verificata passo dopo passo1
Start with the integral \( I_n = \int (\ln x)^n \, dx \), where \( n \) is a positive integer.
Apply integration by parts by choosing \( u = (\ln x)^n \) and \( dv = dx \). Then, compute \( du \) and \( v \):
- \( du = n (\ln x)^{n-1} \cdot \frac{1}{x} \, dx \)
- \( v = x \)
Use the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Substitute the expressions for \( u, v, du \) to get:
\[ I_n = x (\ln x)^n - \int x \cdot n (\ln x)^{n-1} \cdot \frac{1}{x} \, dx \]
Simplify the integral inside:
\[ I_n = x (\ln x)^n - n \int (\ln x)^{n-1} \, dx \]
This gives the reduction formula:
\[ \int (\ln x)^n \, dx = x (\ln x)^n - n \int (\ln x)^{n-1} \, dx \]
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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 is crucial to simplify the integral effectively.
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Integration by Parts for Definite Integrals
Reduction Formula
A reduction formula expresses an integral involving a power or parameter n in terms of a similar integral with a lower power or parameter (n-1). It helps solve complex integrals recursively by breaking them down into simpler cases.
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5:59Recursive Formulas
Logarithmic Functions and Their Derivatives
Understanding the properties and derivatives of logarithmic functions, especially ln(x), is essential. The derivative of ln(x) is 1/x, which plays a key role when applying integration by parts to integrals involving powers of ln(x).
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05:18Derivative of the Natural Logarithmic Function
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